RAPID PATH-PLANNING ALGORITHMS FOR AUTONOMOUS PROXIMITYOPERATIONS OF SATELLITES

By

JOSUE DAVID MUNOZ

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2011

© 2011 Josue David Munoz

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To my dear mother and father

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ACKNOWLEDGMENTS

I would first like to thank my advisor Dr. Norman G. Fitz-Coy, who I have had the

honor of working with since my undergraduate career (and now seems so long ago).

He has continually challenged me throughout the years and has also served as a

role-model on many accounts. His passion for teaching and research has motivated

me to pursue this degree, and his conscientious demeanor has made it a worthwhile

experience.

I would like to express my thanks to my supervisory committee, Dr. Warren E.

Dixon, Dr. Anil V. Rao, and Dr. Thomas F. Burks, who are some of the most brilliant

minds I have had the pleasure of meeting. The knowledge they have instilled on me was

pivotal for completing this work, and for that I am grateful.

I would also like to thank the agencies that have provided me with research

and/or financial support throughout the years. Particularly, the University of Florida,

the Defense Advanced Research Agency Projects, the South East Alliance for Graduate

Education of the Professoriate Fellowship, the Lockheed Martin Corporation, the Air

Force Research Lab/Space Vehicles Directorate, and the Science, Mathematics, and

Research Transformation Scholarship.

Finally, I would like to thank my family, friends, and colleagues who have provided

me with morale and/or research support throughout the years. Particularly, my brothers

and sisters who have educated me in many ways, whether they realize it or not. Thanks

go out to Shawn Johnson, who I met during my first week of college. Without his help,

I would not have the study habits that allowed me to get this far in my career. Thanks

to Ernesto del Castillo, who was always there to lift my spirits when I felt discouraged.

Thanks to Dr. Nick Martinson, who I was able to brainstorm with and was always

available to discuss research ideas. Thanks to Dr. Fred Leve, who always provided

a different perspective, and our conversations always provoked new and interesting

research ideas. Thanks to Dr. George Boyarko, who was always able to put things into

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perspective, and helped bring a lot of my research together. And thanks to my friends

and colleagues Shawn Allgeier, Sharanabasaweshwara Asundi, Dante Buckley, Katie

Cason, Takashi Hiramatsu, Tzu-Yu Lin, Vivek Nagabhushan, Kunal Patankar, and Bungo

Shiotani for the many long hours spent working together on our research and academic

endeavors.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.1 Previous Missions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.2.1 Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . 271.2.2 Analytic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3 Technical Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.4 Research Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 SYSTEM DYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.1 Orbital Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.1.1 Zonal Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.1.2 Aerodynamic Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1.3 Solar Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1.4 Third Body Disturbances . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.1 Gravity Gradient Torque . . . . . . . . . . . . . . . . . . . . . . . . 392.2.2 Aerodynamic Torque . . . . . . . . . . . . . . . . . . . . . . . . . . 392.2.3 Solar Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 Attitude Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3.1 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3.2 Axis-Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.3 Unit Quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 SYSTEM MODELING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1 Target/Chaser Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Actuator Dynamics and Model . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.1 Reaction Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.2 Reaction Wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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4 OPTIMAL TRAJECTORIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1 Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1.1 Minimum Time Problem . . . . . . . . . . . . . . . . . . . . . . . . 574.1.2 Fixed Time Minimum Control Effort Problem . . . . . . . . . . . . . 594.1.3 Finite Horizon Linear Quadratic Problem . . . . . . . . . . . . . . . 60

4.2 Methods for Solving Optimal Control Problems . . . . . . . . . . . . . . . 634.2.1 Indirect Method: Shooting . . . . . . . . . . . . . . . . . . . . . . . 644.2.2 Direct Method: Collocation . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Optimal Rendezvous Trajectories . . . . . . . . . . . . . . . . . . . . . . . 684.3.1 Minimum Time Rendezvous . . . . . . . . . . . . . . . . . . . . . . 694.3.2 Fixed Time Minimum Control Effort Rendezvous . . . . . . . . . . 694.3.3 Finite Horizon Quadratic Cost Rendezvous . . . . . . . . . . . . . 714.3.4 Constrained Fixed Time Minimum Fuel Rendezvous . . . . . . . . 72

4.4 Optimal Slew Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.4.1 Minimum Time Slew . . . . . . . . . . . . . . . . . . . . . . . . . . 764.4.2 Fixed Time Minimum Control Effort Slew . . . . . . . . . . . . . . . 764.4.3 Finite Horizon Quadratic Cost Slew . . . . . . . . . . . . . . . . . . 78

5 ARTIFICIAL POTENTIAL FUNCTION METHOD . . . . . . . . . . . . . . . . . 81

5.1 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.1 Clohessy-Wiltshire-Hill Example . . . . . . . . . . . . . . . . . . . . 835.2.2 Small Angle Approximation Example . . . . . . . . . . . . . . . . . 86

6 ADAPTIVE ARTIFICIAL POTENTIAL FUNCTION METHOD . . . . . . . . . . . 89

6.1 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3.1 Clohessy-Wiltshire-Hill Example . . . . . . . . . . . . . . . . . . . . 976.3.2 Small Angle Approximation Example . . . . . . . . . . . . . . . . . 100

7 FINITE HORIZON LINEAR QUADRATIC PROBLEMS . . . . . . . . . . . . . . 104

7.1 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047.2 Linear Time-Invariant System . . . . . . . . . . . . . . . . . . . . . . . . . 1067.3 Linear Time-Varying System . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.3.1 Picard Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.3.2 Homotopy Continuation . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.4.1 Picard Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.4.2 Homotopy Continuation . . . . . . . . . . . . . . . . . . . . . . . . 113

7.5 Yamanaka-Ankerson-Tschauner-Hempel Example . . . . . . . . . . . . . 116

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8 NUMERICAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.1 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228.2 High Fidelity Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.2.1 Minimum Time Trajectories . . . . . . . . . . . . . . . . . . . . . . 1298.2.2 Fixed Time Minimum Control Effort Trajectories . . . . . . . . . . . 1328.2.3 Finite Horizon Quadratic Cost Trajectories . . . . . . . . . . . . . . 1368.2.4 Disturbance-Free Trajectories . . . . . . . . . . . . . . . . . . . . . 1408.2.5 Artificial Potential Function Trajectory . . . . . . . . . . . . . . . . . 1438.2.6 Adaptive Artificial Potential Function Trajectory . . . . . . . . . . . 1438.2.7 Obstacle Avoidance Trajectories . . . . . . . . . . . . . . . . . . . 1468.2.8 Final Approach Trajectory . . . . . . . . . . . . . . . . . . . . . . . 154

9 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

APPENDIX

A RELATIVE MOTION MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

A.1 Relative Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160A.1.1 Clohessy-Wiltshire-Hill Equation . . . . . . . . . . . . . . . . . . . 163A.1.2 Yamanaka-Ankerson-Tschauner-Hempel Equation . . . . . . . . . 164

A.2 Relative Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

B TWO POINT BOUNDARY VALUE PROBLEM SOLUTION . . . . . . . . . . . . 169

C HOMOTOPY WITH RICCATI EQUATION - L(S) = S . . . . . . . . . . . . . . . 174

D HOMOTOPY WITH TRANSFORMED RICCATI EQUATION - L(X) = X . . . . . 179

E HOMOTOPY WITH TRANSFORMED RICCATI EQUATION - L(X) = X− X . . 183

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

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LIST OF TABLES

Table page

3-1 Reaction wheel system parameters and constraints . . . . . . . . . . . . . . . 54

4-1 Optimal rendezvous parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4-2 Optimal slew parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5-1 Relative translation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5-2 Relative orientation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6-1 Relative translation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6-2 Relative orientation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7-1 Finite horizon linear quadratic parameters . . . . . . . . . . . . . . . . . . . . . 110

7-2 Maximum difference of Riccati matrix components (PI) . . . . . . . . . . . . . . 111

7-3 Costs for each approximating method . . . . . . . . . . . . . . . . . . . . . . . 113

7-4 Maximum difference of Riccati matrix components (HC case 1) . . . . . . . . . 113

7-5 Maximum difference of Riccati matrix components (HC case 2) . . . . . . . . . 114

7-6 Maximum difference of Riccati matrix components (HC case 3) . . . . . . . . . 116

7-7 Final approach parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7-8 Costs for final approach using HC . . . . . . . . . . . . . . . . . . . . . . . . . 120

8-1 Monte Carlo simulation parameters (CWH equation) . . . . . . . . . . . . . . . 123

8-2 Monte Carlo simulation parameters (SAA equation) . . . . . . . . . . . . . . . 124

8-3 Target spacecraft parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8-4 Chaser spacecraft parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8-5 APF method parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8-6 AAPF method parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8-7 Obstacle spacecraft parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8-8 Constrained APF method parameters . . . . . . . . . . . . . . . . . . . . . . . 149

8-9 Constrained AAPF method parameters . . . . . . . . . . . . . . . . . . . . . . 150

8-10 Final approach parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

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B-1 Relative orientation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

B-2 Relative orientation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

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LIST OF FIGURES

Figure page

1-1 Illustration of JAXA’s flight experiments . . . . . . . . . . . . . . . . . . . . . . 22

1-2 Illustration of DART mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1-3 Illustration of the XSS missions . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1-4 Illustration of OEDS mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1-5 Illustration of Prisma mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1-6 Computational efficiency vs. optimality tradeoff . . . . . . . . . . . . . . . . . . 30

2-1 Surface discretization of spacecraft with unit normal vectors . . . . . . . . . . . 36

3-1 High level Simulink diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3-2 Geometry of spacecraft and reaction jets . . . . . . . . . . . . . . . . . . . . . 46

3-3 Simulink satellite model block . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3-4 Thrust profile of reaction jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3-5 Block diagram to achieve thrust profile . . . . . . . . . . . . . . . . . . . . . . . 50

3-6 Illustration of saturation and deadband limits . . . . . . . . . . . . . . . . . . . 50

3-7 Switch to impose saturation and deadband limits . . . . . . . . . . . . . . . . . 51

4-1 Tradeoff between final time and control effort . . . . . . . . . . . . . . . . . . . 61

4-2 Minimum time rendezvous results . . . . . . . . . . . . . . . . . . . . . . . . . 70

4-3 Fixed time minimum control effort rendezvous results . . . . . . . . . . . . . . 71

4-4 Finite horizon quadratic cost rendezvous results . . . . . . . . . . . . . . . . . 73

4-5 Constrained fixed time minimum control effort trajectories . . . . . . . . . . . . 74

4-6 Constrained fixed time minimum control effort rendezvous results . . . . . . . . 75

4-7 Minimum time slew results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4-8 Fixed time minimum control effort slew results . . . . . . . . . . . . . . . . . . . 78

4-9 Finite horizon quadratic cost slew results . . . . . . . . . . . . . . . . . . . . . 80

5-1 Artificial potential function (APF) example . . . . . . . . . . . . . . . . . . . . . 82

5-2 APF method results using Clohessy-Wiltshire-Hill (CWH) equation . . . . . . . 85

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5-3 APF method running cost using CWH equation . . . . . . . . . . . . . . . . . . 86

5-4 APF method results using small angle approximation (SAA) equation . . . . . . 88

6-1 Illustration of adaptive estimate trajectory . . . . . . . . . . . . . . . . . . . . . 94

6-2 Examples where APF does not guarantee convergence . . . . . . . . . . . . . 97

6-3 Adaptive artificial potential function (AAPF) method results using CWH equation 99

6-4 AAPF running cost using CWH equation . . . . . . . . . . . . . . . . . . . . . . 100

6-5 AAPF adaptive estimates using CWH equation . . . . . . . . . . . . . . . . . . 100

6-6 AAPF results using SAA equation . . . . . . . . . . . . . . . . . . . . . . . . . 102

6-7 AAPF adaptive estimates using SAA equation . . . . . . . . . . . . . . . . . . 102

7-1 Picard Iteration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7-2 Homotopy continuation (case 1) results (h = −1/20) . . . . . . . . . . . . . . . 115

7-3 Homotopy continuation (case 2) results (h = −1/2) . . . . . . . . . . . . . . . . 117

7-4 Homotopy continuation (case 3) results (h = −1/2) . . . . . . . . . . . . . . . . 118

7-5 Homotopy continuation results for final approach example (h = −1/10) . . . . . 120

8-1 Initial relative position/orientations used in Monte Carlo simulations . . . . . . . 125

8-2 Costs and settling times data from Monte Carlo simulations . . . . . . . . . . . 126

8-3 Minimum time “trajectory tracking” results . . . . . . . . . . . . . . . . . . . . . 131

8-4 Minimum time “force tracking” results . . . . . . . . . . . . . . . . . . . . . . . . 133

8-5 Fixed time minimum control effort “trajectory tracking” results . . . . . . . . . . 135

8-6 Fixed time minimum control effort “force tracking” results . . . . . . . . . . . . . 137

8-7 Finite horizon quadratic cost “trajectory tracking” results . . . . . . . . . . . . . 139

8-8 Finite horizon quadratic cost “force tracking” results . . . . . . . . . . . . . . . 141

8-9 Fixed time minimum control effort “trajectory tracking” results (no disturbances) 142

8-10 Finite horizon quadratic cost “force tracking” results (no disturbances) . . . . . 144

8-11 APF method results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8-12 AAPF method results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8-13 Constrained fixed time minimum control effort trajectories . . . . . . . . . . . . 148

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8-14 Constrained fixed time minimum control effort “trajectory tracking” results . . . 149

8-15 Constrained APF method trajectories . . . . . . . . . . . . . . . . . . . . . . . 151

8-16 Constrained APF method results . . . . . . . . . . . . . . . . . . . . . . . . . . 152

8-17 Constrained AAPF method trajectories . . . . . . . . . . . . . . . . . . . . . . . 153

8-18 Constrained AAPF method results . . . . . . . . . . . . . . . . . . . . . . . . . 154

8-19 Final approach results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

A-1 Relative orientation between chaser and target . . . . . . . . . . . . . . . . . . 167

B-1 Two point boundary value problem examples . . . . . . . . . . . . . . . . . . . 173

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LIST OF ABBREVIATIONS

AAPF Adaptive artificial potential function

ACL Autonomous control level

ANGELS Autonomous Nanosatellite Guardian for Evaluating Local Space

APF Artificial potential function

APO Autonomous proximity operation

CDM Cell decomposition method

CM Center of mass

CWH Clohessy-Wiltshire-Hill

DART Demonstration of Autonomous Rendezvous Technology

DOF Degrees-of-freedom

DRE Differential Riccati equation

ETS Engineering Test Satellite

FREND Front-end Robotic Enabling Near-term Demonstration

G&C Guidance and control

GAS Globally asymptotically stable

HC Homotopy continuation

HTV H-II transfer vehicle

ISS International Space Station

KKT Karush-Kuhn-Tucker

LCF Lyapunov candidate function

LEO Low Earth orbit

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LQ Linear quadratic

LTI Linear time invariant

LTV Linear time varying

MCS Monte Carlo simulation

MSE Maclaurin series expansion

NLP Nonlinear program

OCM Orthogonal collocation method

OCP Optimal control problem

OEDS Orbital Express Demonstration System

PI Picard iteration

PID Proportional-integral-derivative

PVT Primer vector theory

SAA Small angle approximation

SSA Space situational awareness

STM State transition matrix

SUMO Satellite for the Universal Modification of Orbits

TPBVP Two point boundary value problem

TSE Taylor series expansion

UAV Unmanned aerial vehicle

XSS Experimental Satellite System

YATH Yamanaka-Ankerson-Tschauner-Hempel

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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

RAPID PATH-PLANNING ALGORITHMS FOR AUTONOMOUS PROXIMITYOPERATIONS OF SATELLITES

By

Josue David Munoz

August 2011

Chair: Norman G. Fitz-CoyMajor: Aerospace Engineering

Autonomous proximity operations (APOs) can be bifurcated into two phases: (i)

close-range rendezvous and (ii) final approach or endgame. For each APO phase,

algorithms capable of real-time path planning provide the greatest ability to react to

“unmodeled” events, thus enabling the highest level of autonomy. This manuscript

explores methodologies for real-time computation of APO trajectories for both APO

phases.

For the close-range rendezvous trajectories, an Adaptive Artificial Potential Function

(AAPF) methodology is developed. The AAPF method is a modification of the Artificial

Potential Function (APF) methodology which has favorable convergence characteristics.

Building on these characteristics, the modification involves embedding the system

dynamics and a performance criterion into the APF formulation resulting in a tunable

system. Near-minimum time and/or near-minimum fuel trajectories are obtained by

selecting the tuning parameter. Monte Carlo simulations are performed to assess the

performance of the AAPF methodology.

For the final approach or endgame trajectories, two methodologies are considered:

a Picard Iteration (PI) and a Homotopy Continuation (HC). Problems in this APO phase

are typically solved as a finite horizon linear quadratic (LQ) problem, which essentially

are solved as a final value problem with a Differential Riccati Equation (DRE). The

PI and HC methods are well known tools for solving differential equations and are

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utilized in this effort to provide solutions to the DRE which are amenable to real-time

implementations; i.e., they provide solutions which are functionals to be evaluated

real-time. Several cases are considered and compared to the classical DRE solution.

17

CHAPTER 1INTRODUCTION

The paradigm of space technology is transitioning to enable autonomous on-orbit

operations. [1, 2] One sign of this transition is the imminent retirement of the Space

Shuttle. The Space Shuttle has carried many satellites and astronauts into space,

which resulted in unprecedented missions and technological advancements. The Space

Shuttle has also provided logistics to valuable space assets which would otherwise have

been decommissioned. [3] The retirement of the Space Shuttle is a result of many years

of high operation and maintenance costs. Autonomous spacecraft present an alternative

for performing proximity operations that were previously done manually. [4] This would

also allow manned space flight missions to focus on other space science missions rather

than mundane resupply missions.

Another example of the transition in space technology is the rapidly increasing

number of objects and debris in orbit. As space technology continues to mature,

it is becoming easier to develop and launch space systems into orbit. In addition,

collisions such as the Chinese ASAT test in 2007, the USA 193 intercept in 2008, and

the Iridium 33/Cosmos 2251 collision in 2009 contribute greatly to orbital debris that

could potentially be harmful to neighboring space assets. [5, 6] Currently, collisions

are avoided by planning a series of maneuvers from a ground station and uplinking

the commands. However, as the number of objects in orbit increases, it is becoming

increasingly difficult to keep track of all objects and to solve for collision avoidance

maneuvers with multiple constraints in a timely manner. The Iridium 33/Cosmos 2251

collision is a prime example of this, where the Iridium 33 satellite was a functional

satellite, yet a collision was not avoided. This example shows why space situational

awareness (SSA) is becoming critical to protect valuable space assets and to perform

successful collision avoidance maneuvers. To supplement the SSA capabilities from

18

ground stations, satellites capable of performing autonomous proximity operations

(APOs) can be used to accurately track objects in space or remove orbital debris. [7]

The responsive space initiative and small satellite technology also require the need

for autonomy. One of the requirements of responsive space systems is autonomy such

that they spacecraft is able to change its orbit or perform orbital stationkeeping onboard.

[8, 9]. Likewise, small satellites or groups of fractionated small satellites can supplement

or replace certain space assets. [10, 11] Small satellites can be used to perform APOs

on other space assets as well. [12, 13]

APOs include (but are not limited to) close-range rendezvous, inspection,

interception, docking, payload transfer, orbital station-keeping, formation flying, and

on-orbit assembly. For the discussions in this manuscript, APOs are generalized and

decomposed into two phases: (i) close-range rendezvous and (ii) final approach or

endgame. Being able to accomplish APOs with the highest level of autonomy is pivotal

for:

• Enabling APOs with spacecraft• Giving heritage to technology developed for APOs• Standardization for missions performing APOs• Reduction in supervision needed for performing certain missions• Increased robustness to communication constraints• Enable servicing of spacecraft in orbit

A distinguishing feature of APOs is “autonomy”. Thus, it is necessary to define what

is meant by autonomy in the context of APOs. A proposed definition for an autonomous

system is a system that “must perform well under significant uncertainties in the plant

and the environment for extended periods of time and it must be able to compensate

for system failures without external intervention.” [14] This definition provides insight,

yet it is too broad for the purpose of APOs. A different definition of autonomy has been

proposed by the artificial intelligence, robotics, and intelligent systems community using

three categorizations of model-based architectures: [15]

19

1. Model-based architecture: Ability to achieve prescribed objectives, allknowledge being in the form of models, as in the model-based architecture.

2. Adaptive model-based architecture: Ability to adapt to major environmentalchanges. This requires knowledge enabling the system to perform structurereconfiguration, i.e., it needs knowledge of structural and behavioralalternatives as can be represented in the system entity.

3. Generative model-based architecture: Ability to develop its own objectives.This requires knowledge to create new models to support the new objectives,that is a modeling methodology.

These three categories are useful for high-level analysis, yet a systematic approach

of characterizing autonomy in space systems is desired. [16, 17] The approach taken

by the Air Force Research Lab to characterize autonomy of Unmanned Aerial Vehicles

(UAVs) uses Autonomous Control Levels (ACLs) ranging between 0-10. [18] The

lowest ACL is Level 0, which corresponds to a vehicle that is remotely piloted and

has no decision making capabilities. The highest ACL is Level 10 and is described

as “Human-like”. The intermediate levels are defined for different ranges of abilities

to “detect and track” other vehicles in close proximity and having “decision making

capabilities on-board” with different amounts of “external supervision”. If a similar

classification of autonomy is used for spacecraft, then being able to track objects in

close proximity and rapidly plan optimized trajectories on-board (with minimal external

supervision or intervention) is the level of autonomy desired for APOs.

On the other hand, performing tasks for proximity operation missions offline tends to

have high operation costs. [19] Planning and scheduling offline is computationally costly

and the additional communication constraints presents problems due to low-bandwidth,

high-latency communication links. [20] An example of autonomous system with these

capabilities is the Mars Exploration Rover which landed successfully on Mars in 2004.

The latency of the communication (ranging from 8 to 42 minutes) required that a large

number of tasks be planned and uplinked intermittently and executed without human

monitoring or confirmation. [21] While the delays and communication constraints

with a spacecraft in orbit are less severe, this example illustrates how scheduling

20

and planning autonomous tasks offline can be cumbersome. The communication

constraints also vary depending on the orbit of the spacecraft. The spacecraft could

be available for communication several times a day at best, while at other times it may

not be accessible for long periods of times. Multiple ground stations can be used to

alleviate the communication constraints, however, there is little freedom in strategically

placing ground stations (due to monetary limits and property rights). On the other hand,

increasing the level of autonomy would reduce the need for constant communication

with a spacecraft.

1.1 Previous Missions

One of the first missions to attempt to demonstrate APOs was JAXA’s Engineering

Test Satellite VII (ETS-VII) in 1998. The ETS-VII is shown in Figure 1-1A and consists

of a passive target satellite (ORIHIME) and an active satellite (HIKOBOSHI); both

initially in a docked state. The main objective was to demonstrate relative approach,

final approach, and docking between HIKOBOSHI and ORIHIME. The first experiment

required that both spacecraft detach, fly in formation at a distance of two meters, and

then demonstrate autonomous docking. [22, 23] The experience gained from this

experiment would later be used on the H-II Transfer Vehicle (HTV) in 2009, which

provides logistics to the International Space Station (ISS). [24] The HTV is shown

in Figure 1-1B. The ETS-VII was also to demonstrate a relative approach trajectory

between the two satellites, however, technical difficulties were experienced due to

attitude deviations. In short, every time an orbital maneuver was executed, the thrusters

would change the orientation of the chaser satellite which did not allow the chaser to

track its predetermined trajectory. In order to alleviate this problem, a modification had to

be made to the flight software. [23] Nevertheless, the lessons learned from the ETS-VII

mission led to the success of the HTV. These two missions demonstrate how critical

it is to perform an APO (particularly rendezvous and docking) with high precision and

collision avoidance, especially when docking with a valued space asset like the ISS.

21

A Artist depiction of ETS-VII B Artist depiction of HTV

Figure 1-1. Illustration of JAXA’s flight experiments

Another mission aimed at testing the ability to perform APOs was the Demonstration

for Autonomous Rendezvous Technology (DART) project in 2005. [25] The main

objective was to demonstrate long-range rendezvous, close-range rendezvous, and

collision avoidance with the decommissioned MUBLCOM satellite which had been in

orbit since 1999. [26] Both the DART spacecraft and MUBLCOM satellite are shown in

Figure 1-2. The DART mission resulted in a mishap, where a “soft collision” between

DART and MUBLCOM occurred after DART exhausted all of its propellant. Propellant

management along with navigation and collision avoidance malfunctions caused the two

spacecraft to collide, however, both spacecraft survived. [27]

Figure 1-2. Illustration of DART mission

22

A concurrent mission with DART was the Experimental Satellite System-11

(XSS-11) in 2005 which followed its predecessor, the XSS-10. The XSS-11 mission was

to further demonstrate capabilities for performing autonomous rendezvous and other

APOs with the upper stage of its Minotaur I launch vehicle. [28, 29] Both XSS-10 and

XSS-11 are depicted in Figure 1-3B and Figure 1-3A, respectively. The XSS-11 mission

was successful in completing rendezvous and 75 natural motion circumnavigations.

The XSS-11 spacecraft also conducted APOs with several US-owned decommissioned

satellites, however, these results are not readily available in the public domain. [12, 29]

A Illustration of XSS-10 B Illustration of XSS-11

Figure 1-3. Illustration of the XSS missions

The Orbital Express Demonstration System (OEDS) mission in 2007 was

successful and a landmark for APOs. This mission was able to demonstrate autonomous

robotic payload transfer and reconfiguration of satellites. [30] The mission consisted of

a servicing satellite (ASTRO) and a serviceable satellite (NextSat); both initially in a

docked state. Both the ASTRO and NextSat satellites are shown in Figure 1-4. ASTRO

was able to successfully perform APOs on NextSat including rendezvous at different

ranges, capture, propellant and hardware transfer, which was a major milestone for

23

autonomous technology in space. However, the APOs were performed at a low level of

autonomy, where several ground-based commands were required for OEDS to complete

an operation. [30, 31]

Figure 1-4. Illustration of OEDS mission

The most recent mission is the Prisma satellites, which were developed by the

Swedish Space Corporation and launched on June 15, 2010. [32–34] The Prisma

satellites consist of an active satellite (MANGO) and a passive satellite (TANGO); both

initially in a docked state. MANGO and TANGO are shown in Figure 1-5. This mission

will attempt to experimentally validate certain algorithms and hardware for APOs. The

main directive is to perform a series of formation flying and rendezvous maneuvers at

different ranges. The spacecraft will validate collision avoidance maneuvers when both

are in proximity of each other. This mission will also give flight heritage to the smallest

thrusters developed to date. [32]

There are future missions like the Satellite for the Universal Modification of

Orbits/Front-end Robotic Enabling Near-term Demonstration (SUMO/FREND), the

Autonomous Nanosatellite Guardian for Evaluating Local Space (ANGELS), and

the most recently proposed venture, ViviSat. [35–37] SUMO/FREND is particularly

interesting since details of the trajectory planning algorithms that will be used have been

published in the public domain. [38] Each spacecraft will carry out specific missions for

24

Figure 1-5. Illustration of Prisma mission

space application purposes, as opposed to testing concepts for APOs. These missions

will also be milestones for determining the most appropriate APO methodologies.

1.2 State of the Art

Despite recent and future missions, the most effective path-planning methodology

for APOs is still undetermined. While the methods used in the missions discussed are

undisclosed, it is clear that the desired level of autonomy has not been demonstrated.

To achieve this level of autonomy, there are different frameworks that have been

suggested. One framework suggests developing faster microprocessors such that

existing algorithms (that would normally be executed offline) would be simplified and

executed on-board. Another framework suggests developing new algorithms that are

capable of being executed in hardware that is readily available. While both frameworks

are valid, verifying and validating either requires flight heritage.

Given the successes and/or shortcomings of the missions discussed in the previous

section, the question of whether rapid path-planning algorithms need to be developed

must to be investigated. While a level of autonomy has been demonstrated to be

feasible for in-space operations, forthcoming demands require that certain missions

have this capability of rapid path-planning for higher levels of autonomy. This is

particularly true with the emergence of small satellite technology and the responsive

space initiative. With both of these technical areas, success is measured from a

25

“cheaper, faster, and gets the job done” perspective. Since small satellite technology

can be relatively “cheaper”, this would be an avenue that could be taken to ultimately

determine the most effective path-planning methods for APOs.

In general, path-planning for the translational motion of spacecraft is referred to

as orbital transfers. For this case, a distinction is made between maneuver planning

and trajectory planning. Maneuver planning refers to a series of control actions for

paths with long transfer times (on the order of the orbital period). The control actions

are typically assumed to be impulsive since the engine burn time is a small fraction of

the transfer time. [39, 40] In addition, the trajectory obtained is not as important as the

terminal conditions of the trajectory. Trajectory planning refers to planning a series of

control actions for paths with short transfer times (on the order of fractions of the orbital

period). For this case, the control actions are not necessarily assumed impulsive and

the trajectory obtained is relevant to the mission. Path-planning for the rotational motion

of spacecraft is typically synonymous with trajectory planning since slew maneuvers

have relatively short transfer times. In this manuscript, path-planning and trajectory

planning are used interchangeably since the transfer times for APOs are small.

To this end, several algorithms have been used/proposed for path-planning of

APOs. These algorithms are bifurcated into two groups: optimization methods and

analytic methods. Optimization methods are methodologies derived from optimal

control theory and historically have been studied to a greater extent. Some examples

of optimization methods include Primer Vector Theory (PVT), cell decomposition

methods (CDM), orthogonal collocation methods (OCM), the Inverse Dynamics in the

Virtual Domain (IDVD) method, the Guidance using Analytic Solution (GAS) method, to

name a few. For analytic methods, the word “analytic” is used in the sense that these

methods have low complexity and the solutions are obtained using a fixed number of

computations. Two prominent analytic methods are the glideslope method and artificial

potential function (APF) method.

26

1.2.1 Optimization Methods

Optimal control theory is the default approach for computing optimal trajectories.

[41–43] Using the calculus of variations, one can mathematically determine the

necessary and sufficient conditions for an optimal trajectory based on the constraints

on the system (i.e., dynamics, time, boundary, path, control effort, etc.). The study of

optimal control theory applied to astrodynamics (particularly orbit transfers) is called

Primer Vector Theory (PVT). [41, 42] The primer vector is nothing more than the costate

associated with the velocity state of the satellite. It turns out that solutions to several

relevant optimal trajectories depend on the solution to the primer vector. Despite its

elegance, PVT (as well as general optimal control problems) still yields a two point

boundary value problem (TPBVP) with both the states and costates, which can be

difficult to solve. [42, 44] The coupled translational and rotational motion has also been

posed as an optimal control problem (OCP) using a variational approach. [38, 45]

Cell decomposition methods attempt to facilitate solving the OCP by discretizing

the state space. As a result, an exhaustive tree search is performed to determine the

optimal trajectory. [46–48] Imposing additional system constraints is fairly easy and can

be done without adding much complexity to the algorithm. However, the computational

cost increases dramatically as the number of states increases. In addition, it is difficult

to determine whether the solution obtained is the global optimizer. Although these

exhaustive search methods have high computational cost, they always converge to a

solution (if the problem is well-posed).

One of the most effective and computationally efficient ways to solve an optimal

control problem is using orthogonal collocation methods (OCM). These methods

essentially transcribe the infinite dimensional OCP to an equivalent finite dimensional

nonlinear program (NLP). [49, 50] Reducing the dimensionality of an OCP using

an OCM greatly reduces the computation needed to obtain a solution to an OCP.

However, convergence times are still indeterminate and the solution is only known at the

27

collocation points. Moreover, it is difficult to determine whether the solution obtained is

the global optimizer of the original OCP.

The IDVD method is similar to an OCM except the devised NLP involves solving

for coefficients of a set of user-defined basis functions. This approach greatly reduces

the dimensionality of the NLP and, in turn, the time needed to compute a solution.

These basis functions are chosen based on known behavior of the dynamical system

and known structure of the optimal control. The trajectories are obtained in the virtual

domain (usually an affine transform of the time domain) and then mapped back to the

time domain. While the IDVD method does not provide an exact solution to an OCP,

the shape of the trajectories obtained are similar to the shape of the trajectories that

would be obtained from solving an OCP (assuming appropriate basis functions are

chosen). It has also been shown that these trajectories are near-optimal based on their

performance index values. [51]

A framework that can be implemented for any optimization method is the receding

horizon approach. [52] This approach attempts to segment the problem by considering

a certain horizon of time rather than solving the OCP for the entire time horizon. As

a result, the OCP is solved sequentially until the terminal conditions are met at the

final time. The intent of this approach is to reduce convergence times. However, since

the problem is being solved sequentially, it is suboptimal according to the “Principle of

Optimality”. [44]

The GAS method is an ad hoc method that reduces the dimensionality of an OCP

by employing the receding horizon framework and performing an optimization only in the

time domain. This method requires that an analytic solution be known for the dynamics

of the system. [53] This approach greatly reduces the dimensionality of the problem.

However, an optimization problem still needs to be solved iteratively. While this method

obtains a solution with relatively small computational load, the end result is suboptimal

due to the receding horizon framework.

28

1.2.2 Analytic Methods

The first (and most widely used) analytic method discussed is the glideslope

algorithm. [54] This method is commonly used since it can easily be implemented

due to its low complexity. The cornerstone of this algorithm is the classic two-impulse

rendezvous solution. [55] This algorithm is effective yet the performance obtained is

suboptimal, since there is no consideration of a cost metric. There is also no collision

avoidance logic in the algorithm which does not make it favorable for APOs.

Another analytic method is the APF method, which can be thought of as an

automated way of computing maneuvers for the glideslope method. The APF method

has also been considered as a single-step solution to a local optimization problem (i.e.,

infinitesimally small receding horizon), since only local gradient information is used to

plan a maneuver. [52] If the artificial potential is defined with certain characteristics, then

these maneuvers have been shown to yield favorable convergence properties. [56–61] A

collision avoidance logic can be included by augmenting the APF with artificial potentials

representing avoidance regions. The APF method, however, does not include system

dynamics nor a performance metric and is thus suboptimal. As a result, the trajectories

generated for APOs are not well defined, which is important when being used on a

conservative system such as a spacecraft.

1.3 Technical Challenges

Certain considerations need to be taken with the algorithm development for

APOs. The priority for any APO should be safety and conservation of the space

assets their resources. Therefore, an algorithm used for APOs should be robust to

“unmodeled” events (i.e., rapidly generate new or corrected trajectories) and have a

collision avoidance logic. In addition, the algorithm should optimize the trajectories to be

cost-effective (i.e., with respect to control effort, power, time, computation, etc.).

A challenge with which one is faced when developing algorithms is the tradeoff

between computational efficiency and optimality illustrated in Figure 1-6. Ideally,

29

an algorithm would have the computational efficiency of an analytic method while

maintaining the performance characteristics from optimization methods. Given that

computational efficiency plays a role in APOs, it is difficult to characterize different

algorithms based on a single performance index. Thus, when considering algorithms

for APOs, a posterior ancillary performance index must be considered which quantifies

the computations performed and the convergence time to obtain a solution. Measuring

floating point operations is not practical for optimization methods since their convergence

times are indeterminate. [52, 62] Moreover, the conservative computation environment

on flight hardware requires that power and system resources required be taken into

account. In addition to the constraints on radiation-hardened hardware, only a fraction

of the computation resources would be available since other flight operations must be

executed as well.

Increased Complexity/Optimality

Increased Computational Efficiency

Analytic Methods

Hybrid Methods

Optimization Methods

Figure 1-6. Computational efficiency vs. optimality tradeoff

The challenge of developing an algorithm that is both optimal and computationally

efficient is daunting. The default solution is deferring to Moore’s Law and standing by

for flight hardware that guarantees real-time execution of existing algorithms. However,

implementing existing algorithms on-board flight hardware would result in excess

computations since trajectories would continuously need to be updated. Using analytic

methodologies is more favorable since trajectories are obtained using a fixed number

30

of computations (i.e., functional evaluations); which is readily implementable with

existing hardware. However, the trajectories obtained can still be optimized and their

robustness needs to be verified. Different avenues for developing algorithms must be

continuously explored to determine the most suitable algorithms for APOs. The need for

accurate absolute and relative navigation is an additional challenge. [63] It is essential

to have continuous knowledge of the states and objects in proximity of the autonomous

spacecraft to ensure autonomy.

1.4 Research Scope

The dynamics (for both translational and rotational motion) of a rigid body orbiting

the Earth are presented in Chapter 2. For both types of motion, the governing equations

are presented along with the environmental disturbances experienced in orbit. Different

parameterizations for describing the orientation of a rigid body are then discussed along

with the kinematics for each parameterization.

The development of a high fidelity simulation environment is discussed in Chapter 3.

This includes the Simulink model based on the dynamic models discussed in Chapter 2.

Detailed actuator models are developed for reaction jets and reaction wheels. The

reaction jets are the linear momentum exchange devices used to effect an orbital

maneuver. Reaction wheels are the angular momentum exchange devices used to

effect a change in orientation. The simulation environment is later used in Chapter 8

to characterize optimal trajectories and to determine how the APF and AAPF methods

perform under higher fidelity dynamics.

Chapter 4 discusses some of the properties associated with the pertinent optimal

trajectories for APOs. Namely, these trajectories are minimum time trajectories,

fixed time minimum control effort trajectories, and finite horizon linear quadratic (LQ)

trajectories. It is also shown in Chapter 4 that solving a finite horizon LQ problem

reduces to solving a final value problem with the Differential Riccati Equation (DRE).

The optimal trajectories for rendezvous and slew maneuvers are computed in Chapter 4.

31

A rendezvous maneuver with a path constraint (representative of an obstacle) is also

computed.

The APF algorithm is discussed in Chapter 5 and the adaptive artificial potential

function (AAPF) method is presented in Chapter 6. The AAPF method, which is a

modification of the APF method, is developed to exploit the computational efficiency

of the APF method and increase its optimality by choosing a time dependent form of

the artificial potentials. A stability analysis is performed for both the APF and AAPF

methods in Chapter 6. Two numerical examples are provided for both the APF and

AAPF methods to demonstrate both algorithms’ effectiveness.

Chapter 7 discusses methods for solving finite horizon LQ OCPs. The solution

for an LQ OCP with a linear time invariant (LTI) Hamiltonian matrix is obtained using a

state transition matrix (STM) representation. Two new methodologies are developed

for obtaining the solution to a LQ OCP with a linear time varying (LTV) Hamiltonian

matrix using: the Picard Iteration (PI) and the Homotopy Continuation (HC). A

numerical example with a LTV system is then solved using the PI and three different

HC mappings. An example representing a final approach scenario is solved using the

Yamanaka-Ankerson-Tschauner-Hempel (YATH) relative motion model using the HC.

Chapter 8 presents the results from the numerical analyses performed. First,

the results of two Monte Carlo simulations are presented to verify that the AAPF has

improved performance and convergence characteristics over the APF method. Next,

the results obtained from the simulations performed using the high fidelity model

are presented. The optimal trajectories obtained in Chapter 4 for both the rotational

and translational problems are tracked in the high fidelity simulation environment to

determine whether the trajectories are feasible and to characterize any performance

degradation. The APF and AAPF methods are also implemented in the high fidelity

simulation environment to determine how these methods perform in a high fidelity

model. A set of results for rendezvous with obstacle avoidance is also presented, where

32

tracking a trajectory and the APF and AAPF methods are implemented. Lastly, a final

approach scenario is simulated by using a finite horizon LQ control law in the high

fidelity model. Finally, Chapter 9 presents conclusions from the analyses done in the

previous chapters.

33

CHAPTER 2SYSTEM DYNAMICS

The dynamics equations governing the coupled six degrees-of-freedom (DOF)

motion of a spacecraft in orbit are discussed in this chapter. The spacecraft is assumed

to be a rigid body, thus the motion of a rigid body can be bifurcated into translational

motion and rotational motion. The dynamics associated with both motions are discussed

along with the environmental disturbances that affect an object in orbit. The dynamics

equations are used in the Chapter 3 to model the spacecraft and its environment and in

Chapter 4 as the dynamics constraint to obtain optimal trajectories.

2.1 Orbital Mechanics

Assuming a spherically homogeneous central body (i.e., Earth), which is significantly

larger than the orbiting body (i.e., spacecraft), the motion of the smaller body with

respect to the larger body can be modeled as

r +µ⊕‖r‖3 r = f + ad , (2–1)

where r is the position of the spacecraft relative to the center of mass (CM) of the Earth,

µ⊕ is the Earth’s gravitational parameter, f is the control action (i.e., specific thrust)

applied by the satellite, and ad is the sum of disturbing accelerations acting on the

satellite. When ad = 0, this is known as the Keplerian model. [39, 40] The coupling

between rotational motion and translational motion is introduced in f and ad since

these terms depend on the orientation of the spacecraft. The disturbing accelerations

are caused by the environment experienced in orbit. In particular, the disturbances

discussed are higher order gravitational effects, atmospheric drag, solar drag, and third

body effects (i.e., gravitational effects from the Moon and Sun in particular).

2.1.1 Zonal Harmonics

Modeling the Earth as a distributed mass system (which is not spherically

homogeneous), the gravity effect of the Earth can be determined as the negative

34

gradient of gravity potentials. Using spherical harmonics (zonal, sectoral, and tesseral)

to section the Earth, the gravity potential of the zonal harmonics (which have the

dominating disturbing effect) is

V (r) =µ⊕‖r‖

[1−

∞∑k=2

Jk

(R⊕‖r‖

)kPk(cos(φgc))

],

where Jk is the empirically determined constant for the k th zonal harmonic, R⊕ is the

equatorial radius of Earth, Pk is the k th order Legendre polynomial, and φgc is the

geocentric latitude of the satellite. [39, 40] The negative gradient of this potential yields

the effective gravitational specific force from the zonal harmonics on the spacecraft.

Note that the potential term outside of the series yields the gravity term for the restricted

two-body model (i.e., spherically homogeneous Earth). The sectoral and tesseral

harmonics are omitted since the zonal harmonics have the largest disturbing effect

(i.e., the spherically homogeneous Earth term and the J2 effect). In fact, the J2 effect

is up to 1000 times greater than the next most dominating effect. [39] The decrease in

gravitational effect can also be seen in the subsequent zonal terms since when k → ∞,

then(R⊕‖r‖

)k→ 0. The J2 effect represented in the Earth-Centered Earth-Fixed (ECEF)

frame is

aJ2 = −3J2µ⊕R2⊕

2r 5

[rI

(1− 5r2K

r2

)rJ

(1− 5r2K

r2

)rK

(3− 5r2K

r2

)]T,

where rI , rJ , and rK are the components of r represented in the ECEF frame. For the

purposes of modeling, the first six zonal harmonics are used (pp. 550-552 of [39]).

2.1.2 Aerodynamic Drag

The surfaces of the spacecraft are discretized into n faces to determine the

aerodynamic drag. For a convex spacecraft structure, the aggregate aerodynamic

drag is

aad = −(

1

2

ρ(h)

m‖vrel‖2

n∑i=1

γiAiCD,i ni · vrel)

vrel , (2–2)

35

where m is the mass of the spacecraft, ρ(h) is the air density as a function of altitude h,

Ai is the area of face i , CD,i is the drag coefficient of face i , ni is the unit vector normal to

face i , and γi is defined below. [39, 64]

γi =

1 if ni · vrel > 0

0 otherwise

An illustration of the vectors ni is shown in Figure 2-1 for a spacecraft with a cube

structure. The coupling between translational and rotational motion is evident in this

disturbance since the orientation of the spacecraft determines the aerodynamic drag.

Likewise, the aerodynamic drag may cause a moment which affects the rotational

motion of the spacecraft. The parameter vrel is the unit vector parallel to vrel , which is the

relative velocity between the satellite and the Earth’s atmosphere. It is assumed that the

atmosphere has the same rotational velocity as the Earth. As a result, vrel is defined as

vrel = r − ω⊕ × r ,

where ω⊕ is the rotation rate of the Earth.

x y

z

n1

n2

n3

n4

n5

n6

Figure 2-1. Surface discretization of spacecraft with unit normal vectors

36

Different models exist for describing the air density as a function of height. The

model used is the Exponential Atmosphere Model, which is simple yet fairly accurate for

altitudes h ≤ 1000 km. Using this model, the air density is defined as

ρ(h) = ρ0 exp

(−h − h0H

),

where h0, ρ0, and H are tabulated parameters and also depend on the the altitude.

Values for these parameters can be found in [39].

2.1.3 Solar Drag

Using the same discretization of the surfaces of the spacecraft, the aggregate solar

drag is

asd = −(SP

m

n∑i=1

ηiAiCR,i ni · rsun)

rsun , (2–3)

where SP = 4.57× 10−6 N/m2 is the mean solar pressure, rsun is the unit vector pointing

from the Sun to the satellite’s CM, CR,i is the coefficient of reflectivity of face i , and ηi is

defined as below. [39, 64]

ηi =

1 if ni · rsun > 0

0 otherwise

The coupling between translational and rotational motion is evident in this disturbance

as well. It should be noted that this disturbance is only active when the spacecraft is not

in eclipse. [39, 64] The condition for determining whether a spacecraft is in eclipse is

sin−1(R⊕‖r‖

)≥ cos−1

(r

‖r‖ · rsun)

.

2.1.4 Third Body Disturbances

Third body disturbances are gravitational effects caused by objects other than the

Earth. In particular, the objects that have the most influence on an Earth spacecraft are

the Sun and the Moon. [39] The gravitational effect of the Sun on a body orbiting the

37

Earth is

a = µ

(R − r

‖R − r‖3 −R‖R‖3

),

where µ is the gravitational parameter of the Sun and R is the position of the Sun

relative to the Earth’s center. The gravitational effect of the Moon on a body orbiting the

Earth is

a$ = µ$

(R$ − r

‖R$ − r‖3 −R$‖R$‖3

),

where µ$ is the gravitational parameter of the Moon and R$ is the position of the moon

relative to the Earth’s center. The models used for determining the position of the Sun

and Moon as a function of time are presented in [39].

2.2 Attitude Dynamics

The rotational motion of a rigid body can be modeled using Euler’s second law.

[64, 65] The angular momentum of a rigid body about its CM is

H = J · ω ,

where J is the centroidal inertia dyadic of the rigid body and ω is the angular velocity

relative to an inertial reference frame. Thus, the time rate of change of the angular

momentum is equal to the sum of external torques

dH

dt= J · ω + ω × J · ω = τ + τd , (2–4)

where τ is the control torque and τd is the sum of external disturbing torques acting

on the satellite. During flight, a spacecraft is subjected to external disturbance torques

which affect its motion. For completeness, the disturbing torques discussed are the

gravity gradient torque, aerodynamic torque, and solar torque. A detailed expose of

these disturbances can be found in [39, 64].

38

2.2.1 Gravity Gradient Torque

A gravity gradient torque is experienced when a body is not symmetric (about any

axis) and does not have a homogeneous mass distribution. The gravity gradient torque

is defined as

τgg = 3µ⊕‖r‖3 c× (J · c) ,

where c is the unit vector in the nadir direction.

2.2.2 Aerodynamic Torque

An aerodynamic torque results from the aerodynamic drag in equation (2–2), which

does not act along the CM of the spacecraft (i.e., the CM and geometric center are

distinct). For these cases, the aerodynamic torque is modeled as

τat = −1

2

ρ(h)

m‖vrel‖2

n∑i=1

γiAiCD,i (ni · vrel)2 (vrel × rcp,i) ,

where rcp,i is the vector from the CM to the center of pressure of face i . Specifically, this

vector is defined as

rcp,i = li ni − rcm ,

where li is the distance from the geometric center to the center of pressure of face i

and rcm is the position of the spacecraft’s CM relative to the geometric center of the

spacecraft. [39, 64]

2.2.3 Solar Torque

Solar wind has an effect similar to that of atmospheric wind. Thus, solar drag

produces a torque when the center of pressure is distinct from the CM For these cases,

the solar torque is modeled as

τst = −SPm

n∑i=1

ηiAiCR,i (ni · rsun)2 (rsun × rcp,i) .

39

2.3 Attitude Representations

Numerous parameterizations exist to describe the orientation of a rigid body.

[66] First, the classical Euler angles are discussed. The Euler angle representation

is used in Appendix A to derive a set of equations for relative rotational motion. Next,

the axis-angle representation and the unit quaternion are discussed. These attitude

representations are intimately related and are used to define relative orientations and

the scalar metric to measure for relative orientation errors.

2.3.1 Euler Angles

The orientation of one coordinate system relative to another can be described

using at most a sequence of three Euler rotations. [66, 67] An Euler rotation is a rotation

about one of the axes that defines the orthonormal basis of the coordinate system. This

rotation is represented by a matrix operation. The matrix that rotates a vector about the

first axis is

C(1, θ) =

1 0 0

0 cos θ sin θ

0 − sin θ cos θ

,

about the second axis is

C(2, θ) =

cos θ 0 − sin θ

0 1 0

sin θ 0 cos θ

,

and about the third axis is

C(3, θ) =

cos θ sin θ 0

− sin θ cos θ 0

0 0 1

.

Consequently, a rotation matrix relating vector representations in frames A and

B is defined by a rotation sequence about the i , j , and k axes with angles φ, θ, and ψ,

40

respectively (as seen in equation (2–5)). [65–67]

CBA = C(k ,ψ)C(j , θ)C(i ,φ) (2–5)

Thus, a vector represented in frame A can now be represented in frame B as

Bv = CBAAv .

The kinematics of the Euler angles can be derived by summing all the rotation rates

(represented in the same coordinate system). [66] This results in the expression

ω = S(φ, θ,ψ)

φ

θ

ψ

,

where the definition of S(φ, θ,ψ) depends on the rotation sequence used.

2.3.2 Axis-Angle

The same rotation matrix can be derived by performing a single rotation about a unit

vector e (i.e., eigenaxis) by an angle θ (i.e., eigenangle), and is defined as

CBA = C(e, θ) = cos θI + (1− cos θ)eeT − sin θe× ,

where e× is the skew operator and is defined for an arbitrary column matrix

a =

[a1 a2 a3

]Tbelow. [65, 66]

a× =

0 −a3 a2

a3 0 −a1−a2 a1 0

41

The kinematics for this parameterization are given below. [66]

θ = e · ω

˙e =1

2

[e×ω − cot

(θ

2

)e×(e×ω)

]This representation is useful since the angle can be used as a scalar metric for the

relative orientation between two orientations.

2.3.3 Unit Quaternion

A unit quaternion is a set of four parameters used to represent the orientation of a

rigid body

q =

εη

, (2–6)

where ε is the vector component and η is the scalar component. The unity constraint

requires that qTq = 1. Unit quaternions are related to the axis-angle representation by

ε = sin

(θ

2

)e

η = cos

(θ

2

).

The relationship between Euler angles and quaternions is not direct and can be found in

[67] or [66].

Given a quaternion that relates coordinate system A to coordinate system B, the

rotation matrix using this quaternion is defined as

CBA(q) = ΞT (q)Ψ(q) ,

42

where Ξ(q) and Ψ(q) are defined below. [66, 68]

Ψ(q) =

ηI− ε×

−εT

Ξ(q) =

ηI + ε×

−εT

A favorable attribute of quaternions is that the inverse relationship between the rotation

matrix and the quaternion is nonsingular. This means that a unique quaternion can be

extracted from the rotation matrix which represents the original orientation. [66] Another

favorable attribute is that the quaternion kinematics are bilinear (in the quaternion and

the angular velocity) and are represented as

q =1

2Ξ(q)ω =

1

2Ω(ω)q , (2–7)

where

Ω(ω) =

−ω× ω

−ωT 0

.

Quaternions are particularly useful since an error quaternion between two

orientations can be defined. To define an error quaternion, the quaternion product

operation is first defined as

q1 ⊗ q2 =

[Ψ(q1) q1

]q2 =

[Ξ(q2) q2

]q1 .

As a result, the quaternion inverse is defined as

q−1 =

−εη

,

43

such that the quaternion product of a quaternion and its inverse is the zero quaternion[0T 1

]T. This way, the error quaternion is

qe = q1 ⊗ q−12 , (2–8)

since the zero quaternion is obtained when q1 and q2 are equivalent. An error angle is

also defined based on the relationship between the axis-angle representation and unit

quaternions

θe = 2 cos−1(ηe) ,

where ηe is the scalar component of the error quaternion. This error angle represents a

scalar metric for the relative orientation between two orientations described by q1 and

q2.

In conclusion, the dynamics associated with both translational and rotational

motion of a spacecraft in orbit are presented in this chapter. The translational motion

includes disturbances from the Earth’s oblateness, aerodynamic and solar drag, and

third body gravitational effects. The rotational motion includes disturbances from the

gravity gradient and aerodynamic and solar drag. Different sets of parameterizations

for representing an orientation are also discussed. The dynamics discussed in this

chapter are used to develop a high fidelity model of a spacecraft. They are also used for

obtaining optimal trajectories (while neglecting disturbances) for rendezvous and slew

maneuvers.

44

CHAPTER 3SYSTEM MODELING

The high fidelity simulation environment developed in Simulink is discussed in this

chapter. The simulation environment is used for characterizing performance of different

algorithms by applying them in the high fidelity model of the spacecraft. The simulation

environment includes models for two spacecraft (i.e., chaser and target), where each

spacecraft is modeled using the dynamics discussed in Chapter 2. The chaser satellite

has thrusters for translational control and reaction wheels for attitude control, where a

model for these devices is also discussed in this chapter. These models are important

since the actuator dynamics determine whether the commands from a controller are

realizable. The high level Simulink diagram of the simulation environment is shown in

Figure 3-1.

Figure 3-1. High level Simulink diagram

3.1 Target/Chaser Plant

The satellite geometry for both the target and chaser is based on a cube structure

with side length l = 1 m. The structure is important because it dictates the effects of

the disturbances experienced. The actuator geometry assumes unidirectional thrusters

on each face that have a line of action along the center of mass (CM) of the spacecraft.

45

Figure 3-2 illustrates the arrangement of the six thrusters in the body frame. The

reaction wheel system is such that the spin axis of each reaction wheel is aligned with

each axis of the body frame.

f1

f2

f3

f4

f5

f6

x y

z

Figure 3-2. Geometry of spacecraft and reaction jets

It should be noted that the mass is not modeled as a variable quantity. In reality, the

mass is variable since the reaction jets expend fuel to produce a force. [69] This extra

degree of freedom does have considerable effect on the dynamics, however, it is difficult

to determine how to model this without having a preliminary design of the spacecraft.

Having a variable mass and inertia matrix affects the magnitude of some disturbing

forces and the attitude dynamics. The reaction jets are also affected since they would

have a limited supply of fuel to burn. [69]

The motion of both spacecraft is modeled using the equations discussed in

Chapter 2. Particularly, the translational motion is modeled using the restricted two-body

model while including disturbances. A model of the Sun and Moon position is given in

[39] (as a function of the Julian Date), which determines the solar drag, solar torque, and

third body gravitational effects. The rotational motion is modeled using Euler’s second

law while including the disturbing torques. As a result, the satellite plant is grouped into

46

one block as shown in Figure 3-3, where the only inputs are the force and torque from

the actuators.

Figure 3-3. Simulink satellite model block

3.2 Actuator Dynamics and Model

The actuators chosen for the chaser are low-thrust reaction jets for orbital

maneuvers and reaction wheels for attitude maneuvers. The dynamics and constraints

for both the reaction jets and the reaction wheels are discussed including any delays,

saturation limits, and/or deadband limits. Modeling actuators assist in determining

whether the commanded actions by a particular controller are realizable.

3.2.1 Reaction Jets

Reaction jets are linear momentum exchange devices that generate the force

required for performing orbital maneuvers. Depending on the configuration of the jets

on the spacecraft, they can impart a force and/or moment. In general, the relationship

between each individual reaction jet force, and the resultant force and moment can be

expressed as F

M

= Lf ,

where F and M are the force and moment (expressed in the body fixed frame),

respectively, L is the configuration matrix that determines each individual reaction

jet’s contribution to the force and moment, and f is a column matrix containing the force

magnitudes from each reaction jet.

47

The configuration of the reaction jets chosen is illustrated in Figure 3-2. This

configuration is chosen for simplicity and to avoid controllability issues. The matrix L for

this configuration matrix is decomposed as

L =

L1

L2

,

where

L1 =

1 −1 0 0 0 0

0 0 1 −1 0 0

0 0 0 0 1 −1

and L2 = 0 since the reaction jet forces act along the CM and do not produce a

moment. Thus, given a commanded force Fcomm, the individual reaction jet forces can be

determined using the min-norm solution

f = LT1 (L1LT1 )−1Fcomm .

To enforce the unidirectional constraint (i.e., reaction jets can only produce positive

forces), the following conditions are enforced on the min-norm solution

f ∗2i−1 =

f2i−1 + |f2i | if f2i−1 ≥ 0 and f2i ≤ 0

0 otherwisefor i = 1, 2, 3

f ∗2i =

f2i + |f2i−1| if f2i ≥ 0 and f2i−1 ≤ 0

0 otherwisefor i = 1, 2, 3 .

Thus, the actual force from the reaction jet solution

f∗ =

[f ∗1 f ∗2 f ∗3 f ∗4 f ∗5 f ∗6

]T

48

is

Fact = L1f∗ .

The inherent delays and dynamics associated with the chemical, electrical, and

mechanical processes of the jets are modeled as well. An example of a typical thrust

profile is shown in Figure 3-4. [64] First, a force is commanded at time t0. However,

due to the processes required for turning on the reaction jet there is a delay and is

instead turned on at time t1. Next, the force from the reaction jet must ramp up to the

commanded value which is achieved at time t2. The same delay is seen when the force

commanded is changed at time t3 yet the change begins at time t4. The thrust then

ramps down to the new commanded value at t5. The values for these delays, growth,

and decay times range from a few milliseconds to hundreds of milliseconds. [64] The

block diagram in Figure 3-5 illustrates how this thrust profile is achieved, where the delay

used is 1 ms and the transfer function used is

G(s) =100

s + 100,

which yields a growth and decay time of about 100 ms.

fi

tt0 t1 t2 t3t4 t5

CommandedActual

Figure 3-4. Thrust profile of reaction jets

49

Delay G(s)fcomm fact

Figure 3-5. Block diagram to achieve thrust profile

A saturation and deadband limit is imposed on each reaction jet as well. Figure 3-6

illustrates a generic commanded signal that contains both these limits and the resulting

achievable thrust profile. In order to achieve this thrust profile, a three-part switch is

used to distinguish between the three different cases and is illustrated in Figure 3-7.

From Figure 3-7, the first case corresponds to the saturation limit being active, the

second case is when the deadband limit is active, and the third case is when neither is

active. Figure 3-7 also shows the delay and the transfer function discussed previously.

The saturation limit and deadband limits used are fmax =√

3 N and fdb = 0.01 N,

respectively.

fi

t

Saturation limit

Actual

Deadband limit

Commanded

Figure 3-6. Illustration of saturation and deadband limits

3.2.2 Reaction Wheels

Reaction wheels are angular momentum exchange devices that distribute angular

momentum of the satellite by spinning up or down the reaction wheels. Initially, these

50

Figure 3-7. Switch to impose saturation and deadband limits

devices contain a portion of the satellite’s angular momentum. Since angular momentum

is conserved, when the reaction wheels are spun up (or down), the angular momentum

of the remaining components of the satellite has to be adjusted to maintain a total

constant value. [64, 70] As a result, the orientation of the satellite changes. In general,

the angular momentum of a reaction wheel device can be expressed as the sum of

angular momenta of the reaction wheels

h =

N∑i=1

C(3,φi)C(2, θi)C(3, δi)

Ifw ,iΩi

0

0

,

where φi is the separation angle, θi is the inclination angle, δi is the gimbal angle, Ifw ,i

is the moment of inertia of the reaction wheel about its spin axis, and Ωi is the angular

velocity of the reaction wheel. The rotation matrices express the angular momentum

of each reaction wheel in the body fixed frame. The only time-varying quantity in this

expression is the angular velocities of the reaction wheels. Thus, the time derivative of

51

the angular momentum of the reaction wheel device is

h =

N∑i=1

C(3,φi)C(2, θi)C(3, δi)

Ifw ,iΩi

0

0

,

where Ωi is the angular acceleration of the reaction wheel which produces the effect of a

torque.

Typically, the geometry of the reaction wheel system used has each reaction

wheel’s spin axis aligned with each principal axis of the principal body fixed frame

(i.e., three reaction wheels). If a redundant wheel is used, it is typically skewed such

that its direction is along the diagonal of the principal body fixed frame directions. The

configuration used in the simulation environment is a three reaction wheel attitude

control system with each reaction wheel’s spin axis aligned with each of the axes of

the body fixed frame. As a result, the angular momentum of the reaction wheel system

expressed in the chaser’s body fixed frame is

h =

Ifw ,1 0 0

0 Ifw ,2 0

0 0 Ifw ,3

Ω1

Ω2

Ω3

(3–1)

⇒ h = IfwΩ .

Likewise, the derivative of the angular momentum is expressed as

h = IfwΩ . (3–2)

To derive the control methodology to realize a commanded torque using a reaction

wheel system, the rotational equations of motion are revisited. The angular momentum

of a spacecraft about its CM is expressed as

H = Jω + h ,

52

where the centroidal inertia matrix J does not include the reaction wheels. Assuming

no external torques are acting on the satellite (i.e., no disturbing torques), then the total

angular momentum is conserved. Thus, the inertial time derivative of the total angular

momentum is

H = Jω + h + ω×(Jω + h) = 0

⇒ Jω + ω×Jω = −h− ω×h . (3–3)

Coulomb and viscous friction and stiction are considerable torques that affect the

performance of the reaction wheels. [64] Stiction is not easily modeled and is ignored

since it only affects the reaction wheels when they are at low spin rates. The friction

model used on the reaction wheels is the sum of coulomb and viscous friction

τfriction = τcsgn(Ω) + τvΩ ,

where τc is the Coulomb friction coefficient and τv is the viscous friction coefficient.

Recalling equation (3–3), the internal torque from the reaction wheels is obtained as

τcomm − τfriction = −h− ω×h .

where τcomm is the desired torque from the reaction wheel system. Given an attitude

controller τcomm and feedback of reaction wheel angular velocities, the reaction

wheels’ angular acceleration that would produce the commanded torque is found

using equation (3–1) and equation (3–2), and is

Ω = I−1fw[−τcomm + τfriction − ω×IfwΩ

].

The parameters and constraints used for the reaction wheel device are shown in

Table 3-1. These include saturation limits placed on the angular velocity and angular

acceleration of the reaction wheels.

53

Table 3-1. Reaction wheel system parameters and constraintsParameter Value UnitsIfw ,i 0.625 kg ·m2τc 7.06× 10−4 N ·mτv 1.16× 10−5 (N ·m)/(rad/s)

Ωmax 100 rad/s2

Ωmax 524 rad/s

In this chapter, a simulation environment developed to model two spacecraft in

close proximity is discussed. Included is a plant model for both the chaser and target

spacecrafts. The coupled six degrees-of-freedom dynamics discussed in Chapter 2

are used for both spacecraft. The dynamics and constraints associated with the

actuators on the chaser are also discussed. This model is later used to determine

whether optimal trajectories are realizable and to characterize differences between

theoretical performance and actual performance. It is also used to determine whether

the algorithms developed using linearized models are still valid in a higher fidelity model.

54

CHAPTER 4OPTIMAL TRAJECTORIES

Solutions to an optimal control problem (OCP) can be used for obtaining optimal

trajectories. In this chapter, a general form of an OCP is presented. The OCPs of

particular interest that are discussed are the minimum time OCP, fixed time minimum

control effort OCP, and finite horizon linear quadratic (LQ) OCP. Minimum time and fixed

time minimum control effort trajectories are typically used for close-range rendezvous.

Finite horizon LQ trajectories are typically used for the final approach or endgame of

a proximity operation. It is shown that solving a finite horizon LQ OCP is equivalent

to solving a final value problem with the Differential Riccati Equation (DRE). Finally,

the three pertinent optimal trajectories for both the translational and rotational motion

are computed. In addition, a fixed time minimum control effort problem is solved which

includes an obstacle.

4.1 Optimal Control Problem

Without loss of generality, the continuous time OCP is posed as

minu ∈ Rm

J = Φ(x(t0), x(tf ), t0, tf ) +

∫ tft0

L(x(t), u(t), t) dt ,

subject to

x(t) = f(x(t), u(t), t)

φ(x(t0), x(tf ), t0, tf ) = 0

c(x(t), u(t), t) ≤ 0 ,

where Φ : Rn × Rn × R × R → R is the cost incurred from the boundary conditions,

L : Rn ×Rm ×R→ R is the Lagrangian, f : Rn ×Rm ×R→ Rn is the derivative constraint

(i.e., dynamics) on x, φ : Rn × Rn × R × R → Rk is the column matrix of constraints at

the boundary conditions, and c : Rn × Rm × R → Rl (l ≤ m) is the column matrix of

state and/or control constraints. [41, 43, 44] It is important to note that the “less than or

55

equal to” in c(x(t), u(t), t) ≤ 0 (and when used with vectors or matrices in the rest of

this chapter) is a component-wise operator.

The calculus of variations is used to determine the necessary and sufficient

conditions to solve an OCP. [41, 44] First, an augmented cost function is defined as

Ja = Φ + νTφ+

∫ tft0

L+ λT f + µTc dt ,

where ν ∈ Rk , λ ∈ Rn, and µ ∈ Rl are the costates associated with the constraints φ, f,

and c, respectively. The costate µ has the property

µ ≥ 0 if c = 0 (4–1)

µ = 0 if c ≤ 0 .

As a result, the Hamiltonian is defined as the integrand of the augmented cost as

H = L+ λT f .

The necessary conditions for optimality are given in equation (4–2) and the transversality

conditions are given in equation (4–3). [43, 44]

x =

(∂H∂λ

)Tλ = −

(∂H∂x

)T(4–2)

0 =

(∂H∂u

)T

λ(t0) +

(∂Φ

∂x(t0)

)T+

(∂φ

∂x(t0)

)Tν = 0 if x(t0) is unspecified

−λ(tf ) +

(∂Φ

∂x(tf )

)T+

(∂φ

∂x(tf )

)Tν = 0 if x(tf ) is unspecified (4–3)

56

−H(t0) +∂Φ

∂t0+

(∂φ

∂t0

)Tν = 0 if t0 is unspecified

H(tf ) +∂Φ

∂tf+

(∂φ

∂tf

)Tν = 0 if tf is unspecified .

Note that there is no explicit condition for the costate associated with the state and/or

control constraints (i.e., µ). This costate is determined by equation (4–1) when the

constraint is inactive (i.e., c < 0) or by equation (4–1) and equation (4–2) simultaneously

when the constraint is active (i.e., c = 0). [43]

When state and/or control constraints are imposed and/or when the Hamiltonian

is affine in the control, the Minimum Principle of Pontryagin (MPP) must be used since

the necessary conditions for optimality cannot be employed. [43, 44] In essence, the

MPP provides an additional necessary condition to determine the optimal control. This

principle is stated as

H(x∗,λ∗, u∗, t∗) ≤ H(x∗,λ∗, u, t∗) ∀ u ∈ U ,

where U = u ∈ Rm | c(x∗, u, t∗) ≤ 0 is the set of admissible control inputs. The “ ∗ ”

superscript here and henceforth denotes the optimal solution of the variable.

Using the calculus of variations approach to solve the OCP increases the

dimensionality by introducing new variables (i.e., costates) for each set of constraints.

[44] Solving for the costates is necessary since they determine the optimal control as

well as unknown boundary conditions. Also, the transversality conditions show that the

boundary conditions for the states and costates are not known at the same boundary,

making it difficult to solve for both states and costates simultaneously.

4.1.1 Minimum Time Problem

The minimum time problem can be stated as

minu ∈ U

J =

∫ tft0

1 dt = tf − t0 , (4–4)

57

subject to

x(t) = a(x(t), t) + B(x(t), t)u(t)

x(t0), x(tf ), t0 are specified

umin ≤ ui ≤ umax for i = 1, 2, ... ,m ,

where a : Rn × R → Rn and B : Rn × R → Rn×m. [44] Note that the dynamics

constraint is affine in the control, which is a fair assumption for most dynamical systems.

It is inherent that the final time be unspecified and that saturation limits be placed on

the control. Otherwise, the solution would yield an infinite control action to reach the

terminal condition immediately. Also, given the initial condition and the control constraint,

the final condition xf must to be reachable from the initial condition x0. [44, 71]

The Hamiltonian for this problem is defined as

H = 1 + λT [a + Bu] .

The MPP must be applied and results in the following condition

λ∗TB(x∗, t∗)u∗ ≤ λ∗TB(x∗, t∗)u . (4–5)

If the matrix B is written as

B(x∗, t∗) =

[b1(x∗, t∗) b2(x∗, t∗) · · · bm(x∗, t∗)

],

where bi ∈ Rn×1 are column matrices, then the product in equation (4–5) can be

expressed as

λ∗TB(x∗, t∗)u =

m∑i=1

λ∗Tbi(x∗, t∗)ui . (4–6)

58

Substituting equation (4–6) into equation (4–5), the optimal control that satisfies the

MPP is given as

u∗i =

umax if λ∗Tbi(x∗, t∗) < 0

umin if λ∗Tbi(x∗, t∗) > 0

undetermined if λ∗Tbi(x∗, t∗) = 0

for i = 1, 2, ... ,m .

Note that this problem results in a “bang-bang” control structure. [43, 44] This

means that the control lies on the boundary of U throughout the trajectory until the

terminal conditions are reached. The optimal control depends on the values of the

costates, thus the costates still need to be determined.

4.1.2 Fixed Time Minimum Control Effort Problem

The fixed time minimum control effort problem can be stated as

minu ∈ U

J =

∫ tft0

‖u(t)‖ dt , (4–7)

subject to

3 (4–8)

Note that the final time is specified in this problem. If the final time was not specified,

then the solution would be to apply an infinitesimal control action over an infinite amount

of time. [44] Again, since there is a constraint on the control, the final condition xf must

be reachable from the initial condition x0. [44, 71]

The Hamiltonian for this problem is defined as

H = ‖u‖1 + λT [a + Bu] ,

The MPP must be applied and results in the following condition

‖u∗‖1 + λ∗TB(x∗, t∗)u∗ ≤ ‖u‖1 + λ∗TB(x∗, t∗)u . (4–9)

59

Using equation (4–6) and the definition of the 1-norm, the right hand side can be written

as

‖u‖1 + λ∗TB(x∗, t∗)u =

m∑i=1

|ui |+ λ∗Tbi(x∗, t∗)ui .

The optimal control that satisfies the MPP in equation (4–9) is obtained as

u∗i =

umax if λ∗Tbi(x∗, t∗) < −1

0 if − 1 < λ∗Tbi(x∗, t∗) < 1

umin if λ∗Tbi(x∗, t∗) > 1

undetermined if λ∗Tbi(x∗, t∗) = ±1

for i = 1, 2, ... ,m .

Note that this problem results in a “bang-off-bang” control structure. [43, 44] An

interesting result that also arises from this problem is the tradeoff between the final

time specified and the total control effort as depicted in Figure 4-1. [44] As noted in

the figure, there is a lower bound on the final time specified due to the dynamics and

control constraints. For linear systems, this bound can be determined and is a function

of the initial conditions. [44] This bound exists for nonlinear systems, but it is not as

easily determined. Another interesting result is if there are no path constraints, then the

solution can be approximated using two impulses (at the initial and final time). [53, 72]

This approximation, along with the tradeoff between the fixed time and control effort, is

used later in the algorithm development in Chapter 6.

4.1.3 Finite Horizon Linear Quadratic Problem

The finite horizon LQ problem can be stated as

minu ∈ U

J =1

2xT (tf )Sf x(tf ) +

1

2

∫ tft0

xT (t)Q(t)x(t) + uT (t)R(t)u(t) dt , (4–10)

60

Control effort

Final time specified

Figure 4-1. Tradeoff between final time and control effort

subject to

x(t) = A(t)x(t) + B(t)u(t)

x(t0), t0, tf are specified ,

where A(t) ∈ Rn×n, B(t) ∈ Rn×m, Sf ∈ Rn×n is symmetric positive-semidefinite,

Q(t) ∈ Rn×n is symmetric positive-semidefinte, and R(t) ∈ Rm×m is symmetric

positive-definite. This LQ problem is entitled “finite horizon” since the final time is

specified and finite, “linear” since the dynamics constraint is linear, and “quadratic”

since the cost function is a quadratic function. It is also required that the pair (A, B) be

controllable and the pair (A, Q) be observable. [43, 44, 71]

The Hamiltonian for this problem is defined as

H =1

2(xTQx + uTRu) + λT (Ax + Bu) .

61

One of the necessary conditions for optimality states(∂H∂u

)T= 0

⇒ Ru∗ + BTλ∗ = 0 ,

and leading to the optimal control law

u∗ = −R−1BTλ∗ .

Note that the MPP did not have to be employed here since there are no control

constraints and the Hamiltonian is not affine in the control. From the transversality

conditions, the boundary condition of the costate at the final time is given by

λ∗(tf ) = Sf xf . (4–11)

The other two necessary conditions for optimality yield

x∗ =

(∂H∂λ

)T= Ax∗ + Bu∗

λ∗ = −(∂H∂x

)T= −Qx∗ − ATλ∗ .

Substituting equation (4–16) for the control and writing in state space form yields the

following system x∗

λ∗

=

A −BR−1BT

−Q −AT

x∗

λ∗

,

x(t0)

λ(tf )

=

x0

Sf xf

, (4–12)

where the state matrix is the Hamiltonian matrix.

Given the transversality condition in equation (4–11), the existence of a continuous

linear mapping that relates the costates to the states is explored. This mapping is

assumed to be of the form

λ∗(t) = S(t)x∗(t) , (4–13)

62

where S(tf ) = Sf . The time derivative of equation (4–13) yields

λ∗ = Sx∗ + Sx∗ . (4–14)

Substituting the expressions for the derivatives in equation (4–12) into equation (4–14)

and using the relationship in equation (4–13) yields

[S + Q− SBR−1BTS + SA + ATS

]x∗ = 0 .

Thus, the nontrivial solution (i.e., x∗ 6= 0) is given by

S + Q− SBR−1BTS + SA + ATS = 0 , S(tf ) = Sf , (4–15)

where this equation is the DRE. The DRE is a nonlinear matrix differential equation and

an analytic solution cannot be determined in general. Also, since it only has a boundary

condition specified at the final time, the DRE needs to be integrated backwards in time.

[43, 68] Solving this equation also allows for the optimal control law to be written in state

feedback form as

u∗ = −R−1BTSx . (4–16)

4.2 Methods for Solving Optimal Control Problems

This section discusses the shooting method which is used for solving an OCP

indirectly (i.e., via the calculus of variations approach). This method requires that the

necessary conditions for optimality be set up, and that an initial guess for the unknown

boundary conditions be guessed. A collocation method which is used for solving an

OCP directly (i.e., without introducing costates) is also discussed. This method entails

performing a transcription of the continuous time OCP to a finite dimensional nonlinear

program (NLP) which makes solving the optimization problem more tractable.

63

4.2.1 Indirect Method: Shooting

Recall that the transversality conditions assert that if the state and/or time is

specified at boundary, then the costate and/or Hamiltonian is not specified at that

boundary. Additionally, note that if the Hamiltonian is not an explicit function of time, then

it is constant since

dHdt

=∂H

∂t+∂H

∂xx +

∂H∂λλ

⇒ dHdt

= 0 +∂H

∂x

(∂H∂λ

)T− ∂H∂λ

(∂H

∂x

)T⇒ dHdt

= 0 .

To describe the process of applying the shooting method, an example is used for a

general Hamiltonian boundary value problem (HBVP) where the initial state and time are

specified and the final state and time are unspecified. A similar process can be applied

if a different set of boundary conditions are known. First, the HBVP is established based

on the necessary conditions for optimality asx

λ

H

=

(∂H∂λ

)T−(∂H∂x

)T0

u=u∗

, (4–17)

with x0 and t0 specified.

The principal step of the shooting method is that a guess is made for the unknown

initial values of λ(t0) and H(t0). Subsequently, equation (4–17) is integrated forward

in time until the values obtained for λ(tf ) and H(tf ) from the transversality conditions

are obtained. If they are not obtained, then a new guess is needed and the process

should be repeated. A root-finding scheme can be employed to make refined guesses

for subsequent iterations. However, if a poor initial guess is made, then it is unlikely that

the shooting method converges. Unfortunately, there is no specific way of choosing a

64

“good” initial guess. In addition, each iteration of the shooting method requires numerical

integration, which becomes computationally costly if multiple iterations are required. [73]

4.2.2 Direct Method: Collocation

Different approaches for collocation have been proposed and studied for solving

OCPs. [49, 50, 74] The foundation of these approaches lies in choosing a finite set of

collocation points and setting up an optimization problem (equivalent to the OCP), where

the constraints of the OCP are enforced at the collocation points. This is done by first

approximating the state at the N collocation points using a function approximation

x(t) ≈N∑i=1

Li(t)x(ti) , (4–18)

where ti are the collocation points, x(ti) is the value of the state at the collocation point,

and Li(t) is the Lagrange polynomial

Li(t) =

N∏j=1j 6=i

t − tjti − tj

.

The collocation points are usually determined as roots of a special kind of polynomial.

One approach is to use the roots of an orthogonal polynomial; this is called an

orthogonal collocation method (OCM). [49, 50]

Differentiating equation (4–18) yields

x(t) ≈N∑i=1

Li(t)x(ti) =

N∑i=1

f(x(ti), u(ti), ti) . (4–19)

The dynamics constraint is now enforced using an algebraic equation at each collocation

point. Next, a matrix is defined where the values of the state (which are still to be

65

determined) are concatenated as

X =

xT (t1)

xT (t2)

...

xT (tN)

.

Similarly, a matrix is defined where the values of the control (which are still to be

determined) are concatenated as

U =

uT (t1)

uT (t2)

...

uT (tN)

,

and the values of the dynamic constraints are concatenated as

F(X, U, t1, t2, ... , tN) =

fT (x(t1), u(t1), t1)

fT (x(t2), u(t2), t2)

...

fT (x(tN), u(tN), tN)

.

Thus, the dynamic constraint at each collocation points is written as

DX = F(X, U, t1, t2, ... , tN) ,

where the matrix D includes the derivatives of the Lagrangian polynomials evaluated at

the collocation points. This matrix is not unique and depends on the collocation scheme

used. [49, 50] The state and/or control constraints are employed in a similar manner,

where the constraint is applied at each collocation point. This means the state and/or

66

control constraints are written as

C(X, U, t1, t2, ... , tN) =

cT (x(t1), u(t1), t1)

cT (x(t2), u(t2), t2)

...

cT (x(tN), u(tN), tN)

≤ 0 .

In addition, the integral part of the cost functional is now approximated using a

quadrature rule as ∫ tft0

L(x, u, t) dt ≈N∑i=1

wiL(x(ti), u(ti), ti) ,

where wi are the quadrature weights. [49, 50, 73] Now, the cost functional and

constraints are all functions of the finite set of collocation points. The result is a finite

dimensional optimization problem (i.e., NLP) of the form

minX∈RN×nU∈RN×m

J = J(X, U)

subject to

DX = F(X, U, t1, t2, ... , tN)

φ(x(t0), x(tN), t0, tN) = 0

C(X, U, t1, t2, ... , tN) ≤ 0 .

The solution to a NLP is such that parameters X and U satisfy the Karush-Kuhn-Tucker

(KKT) conditions. [75] However, since the KKT conditions are only enforced at the

collocation points, global optimality (with respect to the original OCP) can not be

guaranteed. Also, collocation methods still require an initial guess at each of the

collocation points, where the convergence time to solve the NLP depends on the quality

of the initial guess. [49, 50, 75]

67

A powerful tool that is available free-of-charge and works in the Matlab environment

in conjunction with the Sparse Nonlinear Optimization (SNOPT) solver is the Gauss

Pseudospectral Optimal Control Software (GPOPS). [76, 77] The GPOPS software

allows the user to easily transcribe a continuous OCP to a NLP and solve the NLP using

SNOPT. GPOPS is used to obtain the optimal trajectories in the following section. These

trajectories are also used in Chapter 8 to determine whether there are performance

differences in a higher fidelity model.

4.3 Optimal Rendezvous Trajectories

Four examples of optimal rendezvous trajectories are presented in this section.

The rendezvous problem involves a passive spacecraft (i.e., target) and an active

spacecraft (i.e., chaser). The translational motion for both spacecraft is governed by the

Keplerian model in equation (2–1) (i.e., no disturbing accelerations). The objective of the

rendezvous problem is

rrel = rc − rt → 0

rrel = rc − rt → 0 ,

where rc and rt are the position of the chaser and target, respectively. The parameters

used are given in Table 4-1. The initial conditions of the target place it in a circular orbit

with a 600 km altitude. The initial conditions of the chaser are chosen such that the

relative position between the chaser and target (in the target’s orbital frame) is

r(t0) =

[500 −500 500

]Tm

and the relative velocity is zero. The following path constraint is also imposed on the

chaser to ensure that it stays in low Earth orbit.

200 km ≤ ‖rc‖ − R⊕ ≤ 1000 km

68

Similarly, the control constraint below is imposed

|fi | ≤ fmax for i = 1, 2, 3 ,

which is representative of a control constraint for reaction jets. The markers in the plots

for each example represent the collocation points used by GPOPS.

Table 4-1. Optimal rendezvous parametersParameter Value Units

rc(t0)[−1182.339411 6816.939420 904.891745

]T kmvc(t0)

[0.175776 −0.963776 7.494102

]T km/srt(t0)

[−1182.959348 6817.396210 904.495486

]T kmvt(t0)

[0.175776 −0.963776 7.494102

]T km/sfmax 0.001 km/s2

4.3.1 Minimum Time Rendezvous

The minimum time problem specified in equation (4–4) requires that both the initial

and final states be specified. The initial states are specified, yet the final states for the

individual spacecraft are free. What is specified, is a relative position and velocity of zero

at the final time. The relative trajectory, relative position, relative velocity, and control

history are shown in Figure 4-2. The relative trajectory shown in Figure 4-2A indicates

how the chaser approaches the target. The relative position and relative velocity plots in

Figure 4-2B and Figure 4-2C, respectively, indicate that the final boundary conditions are

satisfied. The control history plot in Figure 4-2D shows that a control structure similar to

“bang-bang” is obtained. Recall that the optimal control is unspecified when the control

is switched. The minimum time (which is also the cost) for rendezvous is t∗f = 49.86 s.

4.3.2 Fixed Time Minimum Control Effort Rendezvous

The minimum control effort rendezvous problem specified in equation (4–7) requires

that both the initial and final states be specified along with a final time. The initial

states are specified, but the final states for the individual spacecraft are free. What

is specified is a relative position and velocity of zero at the final time. The final time

69

0

0.2

0.4

0.6

0.8

−0.6

−0.4

−0.2

0

0.2

0

0.1

0.2

0.3

0.4

rrel,x (km)rrel,y (km)

rrel,z(k

m)

A Relative position trajectory

0 10 20 30 40

−0.5

0

0.5

rrel,x(k

m)

0 10 20 30 40

−0.5

0

0.5

rrel,y(k

m)

0 10 20 30 40

−0.5

0

0.5

t (s)

rrel,z(k

m)

B Relative position history

0 10 20 30 40−0.05

0

0.05

rrel,x(k

m/s)

0 10 20 30 40−0.05

0

0.05

rrel,y(k

m/s)

0 10 20 30 40−0.05

0

0.05

t (s)

rrel,z(k

m/s)

C Relative velocity history

0 10 20 30 40

−1

0

1

x 10−3

fx(k

m/s2)

0 10 20 30 40

−1

0

1

x 10−3

fy(k

m/s2)

0 10 20 30 40

−1

0

1

x 10−3

t (s)

fz(k

m/s2)

D Optimal control history

Figure 4-2. Minimum time rendezvous results

chosen for this problem is tf = 500 s. The relative trajectory, relative position, relative

velocity, and control history are shown in Figure 4-3. The relative trajectory shown in

Figure 4-3A indicates how the chaser approaches the target. The relative position and

relative velocity plots in Figure 4-3B and Figure 4-3C, respectively, indicate that the final

boundary conditions are satisfied at the final time. The control history plot in Figure 4-3D

shows that a similar control structure to “bang-off-bang” is obtained. Recall that the

optimal control is unspecified when the control switches. The cost obtained (or total

control effort) for this problem is J = 6.61 m/s. It should be noted that this problem

70

required less collocation points since the time range was longer and the optimal control

was zero for a majority of the collocation points.

−0.20

0.20.4

0.60.8

−0.6

−0.4

−0.2

0

0.2

0

0.1

0.2

0.3

0.4

rrel,x (km)rrel,y (km)

rrel,z(k

m)

A Relative position trajectory

0 100 200 300 400 500

−0.5

0

0.5

rrel,x(k

m)

0 100 200 300 400 500

−0.5

0

0.5

rrel,y(k

m)

0 100 200 300 400 500

−0.5

0

0.5

t (s)

rrel,z(k

m)

B Relative position history

0 100 200 300 400 500−2

0

2x 10

−3

rrel,x(k

m/s)

0 100 200 300 400 500−2

0

2x 10

−3

rrel,y(k

m/s)

0 100 200 300 400 500−2

0

2x 10

−3

t (s)

rrel,z(k

m/s)

C Relative velocity history

0 100 200 300 400 500−5

0

5x 10

−4

fx(k

m/s2)

0 100 200 300 400 500−5

0

5x 10

−4

fy(k

m/s2)

0 100 200 300 400 500−5

0

5x 10

−4

t (s)

fz(k

m/s2)

D Optimal control history

Figure 4-3. Fixed time minimum control effort rendezvous results

4.3.3 Finite Horizon Quadratic Cost Rendezvous

The quadratic problem specified in equation (4–10) requires that only the initial

states be specified along with a final time. Instead, a cost is added for the initial and

terminal boundary conditions desired. The time horizon chosen is the same as the

minimum fuel problem (i.e., tf = 500 s). The values used for the matrices that define the

71

cost functional are

Sf = Q =

104 × I 0

0 102 × I

∈ R6×6

R = 10−1 × I ∈ R3×3 .

The relative trajectory, relative position, relative velocity, and control history are

shown in Figure 4-4. The relative trajectory shown in Figure 4-4A indicates how

the chaser approaches the target. The relative position and relative velocity plots in

Figure 4-4B and Figure 4-4C, respectively, indicate that the desired final states are

virtually obtained at the final time. The control history plot in Figure 4-4D shows that the

control decays as the terminal conditions are approached.

4.3.4 Constrained Fixed Time Minimum Fuel Rendezvous

The same cost is used as the fixed time minimum control effort problem with the

added path constraint of

‖rc(t)− ro(t)‖ ≥ 100 m ,

where ro(t) is the position of an obstacle and is governed by the two body relative

motion model in equation (2–1) (neglecting disturbances). The initial conditions used for

the obstacle are

ro(t0) =

[−1182.649497 6817.167778 904.693252

]Tkm

vo(t0) =

[0.175775 −0.963776 7.494102

]Tkm/s .

The relative position trajectories of the chaser are shown at different instances

in time in Figure 4-5. These figures indicate how the chaser is able to avoid the

obstacle and reach the target. The relative position, relative velocity, and control

history are shown in Figure 4-6. The relative position and relative velocity plots shown

in Figure 4-6A and Figure 4-6B, respectively, indicate that the final boundary conditions

72

0

0.2

0.4

0.6

0.8

−0.8

−0.6

−0.4

−0.2

0

0

0.1

0.2

0.3

0.4

rrel,x (km)rrel,y (km)

rrel,z(k

m)

A Relative position trajectory

0 100 200 300 400 500

−0.5

0

0.5

rrel,x(k

m)

0 100 200 300 400 500

−0.5

0

0.5

rrel,y(k

m)

0 100 200 300 400 500

−0.5

0

0.5

t (s)

rrel,z(k

m)

B Relative position history

0 100 200 300 400 500−5

0

5x 10

−3

rrel,x(k

m/s)

0 100 200 300 400 500−5

0

5x 10

−3

rrel,y(k

m/s)

0 100 200 300 400 500−5

0

5x 10

−3

t (s)

rrel,z(k

m/s)

C Relative velocity history

0 100 200 300 400 500

−2

0

2

x 10−4

fx(k

m/s2)

0 100 200 300 400 500

−2

0

2

x 10−4

fy(k

m/s2)

0 100 200 300 400 500

−2

0

2

x 10−4

t (s)

fz(k

m/s2)

D Optimal control history

Figure 4-4. Finite horizon quadratic cost rendezvous results

are satisfied at the final time. The control history in Figure 4-6C that a “bang-off-bang”

control structure is obtained, with a midcourse correction in the “z” direction.

4.4 Optimal Slew Maneuvers

Three examples of optimal slew maneuvers are presented in this section.

Neglecting disturbances, the equations of motion used are the quaternion kinematics in

equation (2–7) and the equations of motion in equation (2–4). The parameters used for

73

0

0.5

1

−0.5−0.4

−0.3−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

rrel,y (km)

Time: 130 sec

rrel,x (km)

rrel,z(k

m)

A t = 130 s

0

0.5

1

0.50.40.30.20.1

0

0.1

0.2

0.3

0.4

0.5

rrel,y (km)

Time: 250 sec

rrel,x (km)

rrel,

z(k

m)

B t = 250 s

0

0.5

1

−0.5−0.4

−0.3−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

rrel,y (km)

Time: 380 sec

rrel,x (km)

rrel,z(k

m)

C t = 380 s

0

0.5

1

−0.5−0.4

−0.3−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

rrel,y (km)

Time: 500 sec

rrel,x (km)

rrel,z(k

m)

D t = 500 s

Figure 4-5. Constrained fixed time minimum control effort trajectories

74

0 100 200 300 400 500

−0.5

0

0.5rrel,x(k

m)

0 100 200 300 400 500

−0.5

0

0.5

rrel,y(k

m)

0 100 200 300 400 500

−0.5

0

0.5

t (s)

rrel,z(k

m)

A Relative position history

0 100 200 300 400 500−2

0

2x 10

−3

rrel,x(k

m/s)

0 100 200 300 400 500−2

0

2x 10

−3

rrel,y(k

m/s)

0 100 200 300 400 500−2

0

2x 10

−3

t (s)

rrel,z(k

m/s)

B Relative velocity history

0 100 200 300 400 500

−2

0

2

x 10−4

fx(k

m/s2)

0 100 200 300 400 500

0

2

4

x 10−4

fy(k

m/s2)

0 100 200 300 400 500

−2

0

2x 10

−4

t (s)

fz(k

m/s2)

C Optimal control history

Figure 4-6. Constrained fixed time minimum control effort rendezvous results

the examples are given in Table 4-2. The objective of the slew maneuver is

ε,ω → 0

η → 1 .

These parameters define a 180 slew maneuver about the first body axis. The control

constraint shown below is enforced

|τi | ≤ τmax for i = 1, 2, 3 ,

75

which is representative of a control constraint for a reaction wheel attitude control

system. The markers in the figures for each example represent the collocation points

used by GPOPS.

Table 4-2. Optimal slew parametersParameter Value Units

J

300 20 1020 100 010 0 200

kg ·m2

q(t0)[1 0 0 0

]T –ω(t0)

[0 0 0

]T rad/sτmax 1 N ·m

4.4.1 Minimum Time Slew

The minimum time problem specified in equation (4–4) requires that the initial and

final states be specified, thus the final states are given by the control objective. The

quaternion, angular velocity, and control history are shown in Figure 4-7. The quaternion

and angular velocity plots in Figure 4-7A and Figure 4-7B, respectively, indicate that the

final boundary conditions are obtained. The control history plot in Figure 4-7C shows

that a control structure similar to “bang-bang” is obtained. The resulting minimum time

(which is also the cost) for the slew maneuver is t∗f = 48.70 s.

4.4.2 Fixed Time Minimum Control Effort Slew

The minimum control effort problem is specified in equation (4–7) had the same

boundary conditions as the minimum time slew maneuver. The final time specified for

this problem is tf = 100 s. The quaternion, angular velocity, and control history are

shown in Figure 4-8. The quaternion and angular velocity plots in Figure 4-8A and

Figure 4-8B, respectively, indicate that the final boundary conditions are obtained at the

final time specified. The control history plot in Figure 4-8C shows that a similar control

structure to “bang-off-bang” is obtained. It is interesting to note that a large number of

collocation points were required, and as a result the solution took a long time to obtain.

This may be due to the fact that the optimal control is unspecified at the switching times.

76

0 10 20 30 40

−1

0

1ǫ 1

0 10 20 30 40

−1

0

1

ǫ 2

0 10 20 30 40

−1

0

1

ǫ 3

0 10 20 30 40

−1

0

1

t (s)

η

A Quaternion history

0 10 20 30 40−0.1

0

0.1

ω1(rad/s)

0 10 20 30 40−0.1

0

0.1

ω2(rad/s)

0 10 20 30 40−0.1

0

0.1

t (s)

ω3(rad/s)

B Angular velocity history

0 10 20 30 40

−1

0

1

τ 1(N

m)

0 10 20 30 40

−1

0

1

τ2(N

m)

0 10 20 30 40

−1

0

1

t (s)

τ 3(N

m)

C Optimal control history

Figure 4-7. Minimum time slew results

The interval where the control is unspecified also has some finite width. The optimal

control solution also had small values in between the initial and final switches in the

control. The final cost (and also the control effort) obtained is J = 24.37 N ·m.

77

0 20 40 60 80 100

−1

0

1ǫ 1

0 20 40 60 80 100

−1

0

1

ǫ2

0 20 40 60 80 100

−1

0

1

ǫ3

0 20 40 60 80 100

−1

0

1

t (s)

η

A Quaternion history

0 20 40 60 80 100−0.04

−0.02

0

0.02

0.04

ω1(rad/s)

0 20 40 60 80 100−0.04

−0.02

0

0.02

0.04

ω2(rad/s)

0 20 40 60 80 100−0.04

−0.02

0

0.02

0.04

t (s)

ω3(rad/s)

B Angular velocity history

0 20 40 60 80 100

−1

0

1

τ1(N

m)

0 20 40 60 80 100−0.5

0

0.5

τ 2(N

m)

0 20 40 60 80 100

−0.5

0

0.5

t (s)

τ 3(N

m)

C Optimal control history

Figure 4-8. Fixed time minimum control effort slew results

4.4.3 Finite Horizon Quadratic Cost Slew

The finite horizon quadratic cost requires that only the initial conditions on the state

and the final time be specified. The cost used for this problem is

J =1

2

[εT (tf )Sf ,1ε(tf ) + sf (1− η(tf ))2 + ωT (tf )Sf ,2ω(tf )

]+ ...

1

2

∫ tft0

εT (t)Q1ε(t) + q(1− η(t))2 + ωT (t)Q2ω(t) + τT (t)Rτ (t)dt ,

78

where the values used for the matrices are

Sf ,1 = Sf ,2 = Q1 = Q2 = I ∈ R3×3

R = 10× I ∈ R3×3

sf = q = 1 .

The final time specified is the same as the minimum control effort problem (i.e., tf =

100 s). The quaternion, angular velocity, and control history are shown in Figure 4-9.

The quaternion and angular velocity plots in Figure 4-9A and Figure 4-9B, respectively,

indicate that the desired final conditions are virtually obtained. The control history plot in

Figure 4-9C indicates that the control is saturated for the initial onset of the maneuver,

and then ramped down. This is attributed to the conservative saturation limits on the

torque given the size of the spacecraft.

In conclusion, the theory related to generating optimal trajectories is presented

in this chapter. In particular, the pertinent OCPs for close-range rendezvous are the

minimum time problem and minimum control effort problem, and the pertinent OCP for

the final approach or endgame of a proximity operation is the finite horizon LQ problem.

A set of trajectories for each OCP is obtained for rendezvous and slew maneuvers.

These trajectories are later used as baseline comparisons for the algorithms developed.

It is also later determined whether these trajectories are realized using a high fidelity

model.

79

0 20 40 60 80 100

−1

0

1

ǫ1

0 20 40 60 80 100

−1

0

1

ǫ 2

0 20 40 60 80 100

−1

0

1

ǫ 3

0 20 40 60 80 100

−1

0

1

t (s)

η

A Quaternion history

0 20 40 60 80 100−0.1

0

0.1

ω1(rad/s)

0 20 40 60 80 100−0.1

0

0.1

ω2(rad/s)

0 20 40 60 80 100−0.1

0

0.1

t (s)

ω3(rad/s)

B Angular velocity history

0 20 40 60 80 100

−1

0

1

τ 1(N

m)

0 20 40 60 80 100

−1

0

1

τ 2(N

m)

0 20 40 60 80 100

−1

0

1

t (s)

τ 3(N

m)

C Optimal control history

Figure 4-9. Finite horizon quadratic cost slew results

80

CHAPTER 5ARTIFICIAL POTENTIAL FUNCTION METHOD

Using an artificial potential is an effective method for deriving guidance laws.

Just as a potential shapes certain force fields (e.g., gravity, electromagnetism, etc.),

an artificial potential can be defined to shape an artificial force field for determining a

control law. The artificial potential must have certain properties such that the resulting

control law provides stable results. This chapter discusses the APF methodology. Two

sets of examples are presented where the APF method is applied to the Clohessy-Wiltshire-Hill

(CWH) equation and the small angle approximation (SAA) equation derived in

Appendix A.

5.1 Development

An artificial potential is defined by superimposing attractive and repulsive potentials.

The purpose of the attractive potential is to impose a global minimum at the desired

terminal position xf . The repulsive potentials create regions of high potential at positions

xi of the state space that are to be avoided. [56–59, 78]

Different forms of artificial potentials have been explored. One form is to use

harmonic functions (i.e., satisfy Laplace’s equation) as potentials since these functions

do no exhibit local minima when superimposed. [79–81] However, the gradient of sinks

and sources (which are typical harmonic functions used for attractive and repulsive

potentials, respectively) is undefined at their center and is near zero away from their

center. Thus, scaling the sinks and sources can be difficult.

Another typical choice for an attractive potential function (and the one used in this

work) is a quadratic function

φa(x) =1

2(x− xf )

TP(x− xf ) , (5–1)

where P is a symmetric positive-definite weighting matrix that shapes the attractive

potential. A typical choice for defining repulsive potentials (and the one used in this

81

work) is using Gaussian functions

φr(x) =

N∑i=1

ψi exp

[−(x− xi)

TNi(x− xi)

σi

],

where ψi and σi are the height and width parameters, respectively, and Ni is a

symmetric positive-definite weighting matrix that shapes the i th repulsive potential.

[57, 59]. Using these functions might yield local minima in the artificial potential

(depending on the parameters chosen for the repulsive potential), which may cause

convergence problems. [82, 83] Regardless of the choice of APFs used, the total

artificial potential is defined as the sum of the attractive and repulsive potentials.

φ = φa + φr (5–2)

An example of an APF consisting of a quadratic attractive potential and two Gaussian

repulsive potentials for a two degree of freedom system is illustrated in Figure 5-1.

A 3D surface plot with contour outline B 2D contour plot

Figure 5-1. Artificial potential function (APF) example

Several combinations of the types of APFs have been used to alleviate some

of the drawbacks associated with each of them. [56, 58, 60, 84–86] Systematically

resolving the local minima problem is beyond the scope of this work. However, since

the close-range rendezvous workspace is sparse, there is a considerable separation

between obstacles and the goal. Thus, local minimum are avoided for most sets of

82

obstacle parameters. If the local minima problem cannot be avoided, then different forms

of potentials should be used.

A feedback controller can be developed from the definition of the APF, where a force

in the opposite direction of the local gradient of the artificial potential is applied when the

rate of change of the artificial potential is positive. This feedback controller is defined as

u(x, x) =

−k∇xφ− (x− xf ) if φ ≥ 0

0 otherwise, (5–3)

where k is a positive gain, and

∇xφ =

(∂φ

∂x

)T⇒∇xφ = P(x− xf )−

N∑i=1

2ψiσi

exp

[−(x− xi)

TNi(x− xi)

σi

]Ni(x− xi)

φ =

(∂φ

∂x

)dx

dt+

(∂φ

∂xf

)dxfdt

+

N∑i=1

(∂φ

∂xi

)dxidt

⇒ φ = (x− xf )TP (x− xf )−

N∑i=1

2ψiσi

(x− xi)T exp

[−(x− xi)

TNi(x− xi)

σi

]Ni (x− xi) .

5.2 Numerical Examples

Two sets of numerical examples are presented to demonstrate the APF method. In

the first example, the APF method is applied to the CWH equation with a grid of 14 static

obstacles in the state space. In the second example, the APF method is applied to the

SAA equation with a tumbling satellite.

5.2.1 Clohessy-Wiltshire-Hill Example

The parameters used in this example are given in Table 5-1. The goal of the chaser

(initially located at the relative position r0) is to reach the target (located at the origin)

while avoiding the obstacles. A set of 14 static obstacles are placed in the state space at

the positions shown below.

83

r1 =

[500 0 0

]Tr2 =

[−500 0 0

]Tr3 =

[0 500 0

]Tr4 =

[0 −500 0

]Tr5 =

[0 0 500

]Tr6 =

[0 0 −500

]Tr7 =

[250 250 250

]Tr8 =

[250 −250 250

]Tr9 =

[250 250 −250

]Tr10 =

[250 −250 −250

]Tr11 =

[−250 250 250

]Tr12 =

[−250 −250 250

]Tr13 =

[−250 250 −250

]Tr14 =

[−250 −250 −250

]TThese positions define an obstacle grid that does not yield local minima, yet still clutters

the state space. Static obstacles are practical for avoiding spacecraft that are tethered

or flying in formation. Otherwise, these obstacles would not be static and would drift.

However, static obstacles are used for simplicity and to demonstrate the collision

avoidance abilities of the APF algorithm.

Table 5-1. Relative translation parametersParameter Value Units

n 0.0012 rad/sr0

[−750 500 −750

]T mv0

[0.5 −2 0

]T m/srf

[0 0 0

]T mfmax 1 m/sP I ∈ R3×3 s−2

Ni I ∈ R3×3 s−2

ψi 1.5× 105 –σi 1.0× 104 m2

k 0.002 –

For this example, the control constraint shown below is enforced.

‖f‖2 ≤ fmax

Settling time is used as a performance metric, where the trajectory obtained is

considered settled when ‖r‖ ≤ 10 m. The running control effort cost is also used as

84

a performance metric and is defined as

V (t) =

∫ tt0

‖u‖1 dε .

The results obtained are displayed in Figure 5-2. The trajectory plot in Figure 5-2A

shows that an obstacle is encountered and the chaser is able to maneuver around it.

The position and velocity plots in Figure 5-2B and Figure 5-2C, respectively, indicate

that the chaser is able to reach the target with zero velocity. The corresponding control

history is given in Figure 5-2D.

−1000

0

1000

−1000−500

0500

1000

−1000

−500

0

500

1000

y (m)x (m)

z(m

)

A 3D plot of trajectory

0 500 1000 1500 2000 2500 3000−1000

0

1000

x(m

)

0 500 1000 1500 2000 2500 3000−500

0

500

y(m

)

0 500 1000 1500 2000 2500 3000−1000

0

1000

t (s)

z(m

)

B Position histories

0 500 1000 1500 2000 2500 3000−2

0

2

vx(m

/s)

0 500 1000 1500 2000 2500 3000−5

0

5

vy(m

/s)

0 500 1000 1500 2000 2500 3000−2

0

2

t (s)

vz(m

/s)

C Velocity histories

0 500 1000 1500 2000 2500 3000−1

0

1

∆vx(m

/s)

0 500 1000 1500 2000 2500 3000−1

0

1

∆vy(m

/s)

0 500 1000 1500 2000 2500 3000−1

0

1

t (s)

∆vz(m

/s)

D Control histories

Figure 5-2. APF method results using Clohessy-Wiltshire-Hill (CWH) equation

85

The settling time obtained is 2620 s and the control effort at the settling time is

16.94 m/s2. The running cost is shown in Figure 5-3 which illustrates how the cost

increases with every control action.

0 500 1000 1500 2000 2500 30000

2

4

6

8

10

12

14

16

18

t (s)

V(t)

Figure 5-3. APF method running cost using CWH equation

5.2.2 Small Angle Approximation Example

The parameters used in this example are shown in Table 5-2. The goal of the

chaser (with initial relative orientation α0 to the target) is to reach a relative orientation

of α = 0 rad with the target that is tumbling at the rate ωt. The control constraint shown

below is enforced.

|τi | ≤ τmax for i = 1, 2, 3

Settling time is used as performance metric where the trajectory obtained is considered

settled when ‖α‖ ≤ 0.01 rad. The running control effort cost is also used as a

performance metric and is defined as

V (t) =

∫ tt0

‖τ‖1 dε .

The results for this example are displayed in Figure 5-4. The relative orientation

and relative angular velocity plots in Figure 5-4A and Figure 5-4B, respectively, indicate

that both these parameters converge to zero. The corresponding control history is

86

Table 5-2. Relative orientation parametersParameter Value Units

J

300 20 1020 100 010 0 200

kg ·m2

ωt[0.25 −0.25 0.25

]T rad/sα0

[0.3 −0.2 −0.15

]T radα0

[0 0 0

]T rad/sαf

[0 0 0

]T radτmax 20 N ·m

P I ∈ R3×3 s−2

k 2 –

given in Figure 5-4C. The settling time obtained is 22.7 s. Note that the running cost in

Figure 5-4D linearly increases with time since the controller continuously compensates

for the tumbling satellite.

In conclusion, the APF method is discussed in this chapter. The procedure for

defining an APF and the corresponding control law is presented. Two examples of

the APF method are also presented using the CWH and SAA equations. From the

discussion and examples in this chapter, the APF method is a useful tool for computing

guidance laws as long as its restrictions are understood.

87

0 20 40 60 80 100−0.5

0

0.5

α1(rad)

0 20 40 60 80 100−0.5

0

0.5

α2(rad)

0 20 40 60 80 100−0.2

0

0.2

t (s)

α3(rad)

A Position histories

0 20 40 60 80 100−0.2

0

0.2

α1(rad/s)

0 20 40 60 80 100−0.2

0

0.2

α2(rad/s)

0 20 40 60 80 100−0.2

0

0.2

t (s)

α3(rad/s)

B Velocity histories

0 20 40 60 80 100−20

0

20

∆α1(rad/s)

0 20 40 60 80 100−20

0

20

∆α

2(rad/s)

0 20 40 60 80 100−20

0

20

t (s)

∆α

3(rad/s)

C Control histories

0 20 40 60 80 1000

500

1000

1500

2000

2500

3000

t (s)

V(t)

D Running cost

Figure 5-4. APF method results using small angle approximation (SAA) equation

88

CHAPTER 6ADAPTIVE ARTIFICIAL POTENTIAL FUNCTION METHOD

The artificial potential function (APF) method discussed in Chapter 5 is a method

that can be used in a rapid path-planning framework. It was shown that the APF method

is well-suited for close-range rendezvous scenarios and relative orientation problems.

However, a drawback of the APF method is that it has no consideration of the dynamics

or a performance index. The adaptive artificial potential function (AAPF) method is

a modification of the APF method with the intent of embedding the dynamics and

performance index in the formulation. The APF framework is adopted because of its

low complexity and favorable convergence characteristics. The development of the

AAPF method and the adaptive update law used to determine the weighting parameters

of the APF is discussed in this chapter. A stability analysis is also presented which

shows that stability is obtained for unconstrained cases (i.e., no obstacle regions),

but that stability cannot be proven for constrained cases (i.e., with obstacle regions).

Finally, two numerical examples are presented where the AAPF method is applied to

the Clohessy-Wiltshire-Hill (CWH) equation and the small angle approximation (SAA)

equation derived in Appendix A.

6.1 Development

It was shown in Chapter 5 that different forms of APFs exist. However, there is no

systematic approach for defining an APF for a particular set of dynamics or performance

criteria. The parameters that define APFs are chosen ad hoc, thus it is difficult to

determine the parameters which yield the best performance with least control effort.

Additionally, since there is no consideration of the dynamics, the trajectories obtained

from the APF method are not defined such that the goal is reached at a specified time.

These factors are particularly important with close-range rendezvous of spacecraft,

where control effort, power, and time must be conserved. Therefore, path-planning

algorithms for close-range rendezvous must take these factors into account. To remedy

89

the drawbacks of using the APF method for close-range rendezvous, the AAPF method

is developed by embedding the dynamics and a performance criteria in the formulation.

To embed the dynamics in the formulation of the AAPF method, the relative motion

models discussed in Appendix A are used. These models are used since the two point

boundary value problem (TPBVP) solutions are obtained analytically and the TPBVP

solution is used in the formulation to include the dynamics. To do this, consider the same

form of the attractive potential in equation (5–1) except let the attractive weighting matrix

be time dependent (i.e., P = P(t)). To enforce the symmetric positive-definite condition,

a Cholesky factorization P(t) = RT (t)R(t) is used,where R(t) is the upper triangular

matrix termed the “Cholesky factor” and is defined as

R(t) =

ρ11(t) ρ12(t) ρ13(t)

0 ρ22(t) ρ23(t)

0 0 ρ33(t)

.

Due to the adaptation, the time derivative of the attractive potential now contains

additional terms as shown below.

φa =∂φa∂t

+

(∂φa∂x

)dx

dt+

(∂φa∂xf

)dxfdt

⇒ φa =1

2(x− xf )

T (RTR + RT R)(x− xf ) + (x− xf )TRTR (x− xf )

A time dependent attractive is used since this is what governs how the goal is approached.

Using a time dependent repulsive potential simply changes the shape and size of the

obstacle region and does not necessarily improve performance. As a result, the form of

the repulsive potential remains the same.

Since the form of the APF is the same, the form of the control law also remains the

same as shown in equation (5–3). The difference is that the weights in R(t) must be

chosen effectively to improve performance. This is achieved by allowing the negative

gradient of the attractive potential to adapt to the velocity profile of the TPBVP solution.

90

The formulation of the TPBVP velocity profile is presented in Appendix B. The velocity

profile obtained in equation (B–6) must then be written in a feedback form by letting the

time t0 = t and the initial condition x0 = x. [40]

xd(x, t) = Φ22(−T )Φ−112 (−T )x + Φ−112 (T ) [xf − xt ] (6–1)

Note that xt is now time dependent and is shown below.

xt =

∫ Tt

Φ12(T − ε)B2u dε

For the relative translation problem, the states in question are the components of

the relative position (i.e., x = r). To obtain the velocity profile, equation (6–1) is used and

is defined as

vd = Φ22(−T )Φ−112 (−T )r −Φ−112 (T )rf ,

where Φ12, Φ22 are blocks of the STM for the CWH equation and T is the transfer time

of the TPBVP solution for the CWH equations. Note that rt = 0 since u = 0 (i.e., no

constant forcing term). Also, note that the velocity profile depends on T , where this

parameter is chosen such that Φ−112 (T ) exists.

The relative orientation problem is similar where the states in question are the

components of the relative orientation (i.e., x = α). To obtain the velocity profile,

equation (6–1) is used and is defined as

αd = Φ22(−T )Φ−112 (−T )α+ Φ−112 (T )αf ,

where Φ12, Φ22 are blocks of the STM for the SAA equation and T is the transfer time

of the TPBVP solution for the SAA equations. Note that αt = 0 which corresponds to

always having a compensation for the constant forcing term u = −ω×t Jωt . Also, note

that the velocity profile depends on the choice of T , where this parameter is chosen

such that Φ−112 (T ) exists. This transfer time and the transfer time of the TPBVP solution

91

for the CWH equations is distinct since the STMs are different. For the remainder of

the derivation, an arbitrary state x is used since the velocity profile for both the CWH

equation and the SAA equation is of the same form.

The adaptive update law is now developed to choose the weights of the attractive

artificial potential. First, an error variable is defined as

e = xd − (−∇xφa)

⇒ e = Φ22(−T )Φ−112 (−T )x + Φ−112 (T )xf + RTR(x− xf ) . (6–2)

Next, the time derivative of equation (6–2) yields

e = Φ22(−T )Φ−112 (−T )x + Φ−112 (T )xf + (RTR + RT R)(x− xf ) + RTR (x− xf ) ,

which is linearly parameterized as

e = Φ22(−T )Φ−112 (−T )x + Φ−112 (T )xf + (Y + Z)θ + RTR(x− xf ) ,

where

Y =

ξ1 0 0 0 0 0

0 ξ1 0 ξ2 0 0

0 0 ξ1 0 ξ2 ξ3

Z =

ρ11(x1 − xf ,1) ρ11(x2 − xf ,2) ρ11(x3 − xf ,3) 0 0 0

ρ12(x1 − xf ,1) ρ12(x2 − xf ,2) ρ12(x3 − xf ,3) ρ22(x2 − xf ,2) ρ22(x3 − xf ,3) 0

ρ13(x1 − xf ,1) ρ13(x2 − xf ,2) ρ13(x3 − xf ,3) ρ23(x2 − xf ,2) ρ23(x3 − xf ,3) ρ33(x3 − xf ,3)

θ =

[ρ11 ρ12 ρ13 ρ22 ρ23 ρ33

]T.

92

The terms ξi for i = 1, 2, 3 are defined below.

ξ1 = ρ11(x1 − xf ,1) + ρ12(x2 − xf ,2) + ρ13(x3 − xf ,3)

ξ2 = ρ22(x2 − xf ,2) + ρ23(x3 − xf ,3)

ξ3 = ρ33(x3 − xf ,3)

An adaptive update law is then defined for the weights as

θ = −(Y + Z)#[Φ22(−T )Φ−112 (−T )x + Φ−112 (T )xf + RTR (x− xf ) + e

], (6–3)

where

(Y + Z)# = (Y + Z)T ((Y + Z)(Y + Z)T )−1

which drives e → 0 as t → ∞. Note that the matrix (Y + Z) is full rank except when

x = xf . However, this only occurs at t = ∞ in the APF framework since the controller

results in x→ xf asymptotically. Therefore, (Y + Z) is nonsingular except at t =∞. This

indicates that this algorithm should only be used for close-range rendezvous or intercept

or until the chaser is within a threshold of xf . In addition, a nonzero initial condition for θ

is needed to initially have a positive-definite weighting matrix.

Since an adaptation is being done with a time-varying signal (i.e, the velocity

profile), a discontinuous projection algorithm is employed to ensure that the adaptive

estimates are bounded. [87, 88] First, if “close proximity” situations are only considered,

then there exists δ ∈ R such that

‖x(t)− xf ‖ ≤ δ ∀ t ∈ [t0,∞) .

As a result, the velocity profile in equation (6–1) is also bounded. Thus, there exists an

upper and lower bound θi , θi ∈ R for i = 1, 2, ... , 6, such that a convex set can be defined

93

for each estimate as

Λi = ν ∈ R | θi ≤ ν ≤ θi for i = 1, 2, ... , 6 .

The projection operator is defined based on the definition of the convex set as

˙θi = Proj(θi) =

θi if θi ∈ Λi

θi if θi = θi and θi ≥ 0

θi if θi = θi and θi ≤ 0

0 otherwise

for i = 1, 2, ... , 6 , (6–4)

where θi for i = 1, 2, ... , 6 is defined in equation (6–3). Figure 6-1 shows an illustration

of the resultant adaptive estimate trajectory using the projection algorithm. Using

the adaptive update law in equation (6–3) with the artificial projection algorithm in

equation (6–4), the AAPF control law is

u(x, x) =

−RTR (x− xf )− k∇xφr − (x− xf ) if φ ≥ 0

0 otherwise, (6–5)

where the matrix R contains the adaptive estimates.

θi

θi

θi

t

θi (t)

Figure 6-1. Illustration of adaptive estimate trajectory

94

6.2 Stability Analysis

To analyze the stability of the APF and AAPF method, the unconstrained case (i.e.,

obstacle-free) is first considered. Consider a Lyapunov candidate function (LCF) that

has the same form as the quadratic attractive potential

V (ε) =1

2εTε ,

where ε = x− xf → 0 is the control objective. Given a LCF, the sufficient conditions for a

globally asymptotically stable (GAS) equilibrium point at ε = 0 are: [89]

1. V (0) = 0

2. V (ε) > 0 ∀ x ∈ Rn − 03. V (ε) is radially unbounded (i.e., V (ε)→∞ as ‖ε‖ → ∞)4. V (ε) < 0 ∀ ε ∈ Rn − 0

The first three conditions are satisfied based on the choice of the LCF. For the fourth

condition, the derivative of the LCF is

V = εT (x− xf ) .

If the control action is impulsive, then the velocity at any instant in time can be described

as the velocity plus the magnitude of the impulse (given the impulse is nonzero at

that instant in time). [71] To ensure the fourth condition is satisfied, an impulsive

control action (i.e., ∆x) is defined according to the control law in equation (5–3) or

equation (6–5) that occurs when φ ≥ 0. Thus, the velocity after the impulsive action

occurs is defined as

x− xf + ∆x = −k∇xφ(ε) = −kPε ,

where k is a positive scalar. [57] Note that ∇xφ = P∇xV . Thus, defining the controller in

equation (5–3) or equation (6–5) ensures the velocity is a scalar multiple of the negative

95

gradient of the APF (and LCF) when φ ≥ 0. The derivative of the LCF then becomes

V =

−kεTPε if φ ≥ 0

εT (x− xf ) if φ < 0

< 0 ,

and satisfies the fourth condition. Therefore, all the conditions are met and the point

ε = 0 is a GAS equilibrium point.

When considering obstacle regions, the same stability analysis cannot be used. For

one, choosing a LCF becomes difficult since the same form of the APF does not satisfy

the first two conditions for a GAS equilibrium point. One way to remedy this is to define

the APF as

φ =1

2(x− xf )

T

[P +

N∑i=1

ψi exp

[−(x− xi)

TNi(x− xi)

σi

]I

](x− xf ) ,

and use a LCF of this form as well. However, this still does not resolve the local minima

problem. [90] In fact, it is known that stability cannot be proven with a general set of

obstacle regions and parameters. [82] Consider a two degrees-of-freedom system

example illustrated in Figure 6-2A. If the obstacles are arranged as illustrated and the

obstacle parameters are defined such that the repulsive potentials are not sensed until

the chaser is near the “barricade” of obstacles, then the chaser would not be able to

escape the “barricade” of obstacles. Another example shown in Figure 6-2B occurs

when the dynamics of the system are negligible (e.g., the double integrator) and the

obstacle is along the line of sight of the target. In this scenario, if the obstacle region is

defined symmetrically and it lies along the line of sight of the chaser to the target, then

the attractive and repulsive potentials eventually counteract each other and the control is

null at this point.

Regrettably, the stability for an obstacle avoidance problem using the APF or AAPF

method is based on intuition. The intuitive nature of the APF method gives a notion that

by defining the APF appropriately, the solution converges to the goal position. The two

96

x1

x2

keep out zones

target

chaser

A Local minimum example

x1

x2

keep out zone

target

chaser

B Counteracting obstacle example

Figure 6-2. Examples where APF does not guarantee convergence

examples demonstrate how the APF method does not yield convergence for an arbitrary

dynamical system with arbitrary artificial potential parameters. These types of problems

can be dealt with in an ad hoc manner where the appropriate parameters are used

such that local minima are avoided. [53, 84] The AAPF method, however, considers

the dynamics which alleviates some of the local minima problems. In addition, the

close-range rendezvous scenario has free-floating target and obstacles which reduces

the probability of local minima as well.

6.3 Numerical Examples

Two sets of numerical examples are presented in this section to demonstrate the

AAPF method. In the first example, the AAPF method is applied to the CWH equation

with a grid of 14 static obstacle in the state space. In the second example, the AAPF

method is applied to the SAA equation with a tumbling satellite.

6.3.1 Clohessy-Wiltshire-Hill Example

The parameters used in this example are given in Table 6-1. The goal of the chaser

(initially located at the relative position r0) is to reach the target (located at the origin)

while avoiding the obstacles. A set of 14 static obstacles are placed in the state space at

the positions shown below.

97

r1 =

[500 0 0

]Tr2 =

[−500 0 0

]Tr3 =

[0 500 0

]Tr4 =

[0 −500 0

]Tr5 =

[0 0 500

]Tr6 =

[0 0 −500

]Tr7 =

[250 250 250

]Tr8 =

[250 −250 250

]Tr9 =

[250 250 −250

]Tr10 =

[250 −250 −250

]Tr11 =

[−250 250 250

]Tr12 =

[−250 −250 250

]Tr13 =

[−250 250 −250

]Tr14 =

[−250 −250 −250

]TFor this example, the control constraint shown below is enforced.

‖f‖ ≤ fmax

Settling time is used as a performance metric where the trajectory obtained is

considered settled when ‖r‖ ≤ 10 m. The running control effort cost is also used as

a performance metric and is defined as

V (t) =

∫ tt0

‖u‖1 dε .

Table 6-1. Relative translation parametersParameter Value Units

n 0.0012 rad/sr0

[750 0 −500

]T mv0

[0.5 2 0

]T m/srf

[0 0 0

]T mfmax 1 m/s2

θ0 0.002×[1 0 0 1 0 1

]T s−2

Ni I ∈ R3×3 s−2

ψi 1.5× 105 –σi 1.0× 104 m2

k 0.002 –T 1000 s

98

The results obtained are displayed in Figure 6-3. The trajectory plot in Figure 6-3A

shows that an obstacle is encountered and the chaser is able to maneuver around it.

The position and velocity plots in Figure 6-3B and Figure 6-3C, respectively, indicate

that the chaser reaches the origin with zero velocity. The corresponding control history is

given in Figure 6-3D.

−1000

0

1000 −1000

−500

0

500

1000

−1000

−500

0

500

1000

y (m)x (m)

z(m

)

A 3D plot of trajectory

0 500 1000 1500 2000 2500 3000−1000

0

1000

x(m

)

0 500 1000 1500 2000 2500 3000−500

0

500

y(m

)

0 500 1000 1500 2000 2500 3000−500

0

500

t (s)

z(m

)

B Position histories

0 500 1000 1500 2000 2500 3000−2

0

2

vx(m

/s)

0 500 1000 1500 2000 2500 3000−2

0

2

vy(m

/s)

0 500 1000 1500 2000 2500 3000−1

0

1

t (sec)

vz(m

/s)

C Velocity histories

0 500 1000 1500 2000 2500 3000−1

0

1

∆vx(m

/s)

0 500 1000 1500 2000 2500 3000−1

0

1

∆vy(m

/s)

0 500 1000 1500 2000 2500 3000−1

0

1

t (sec)

∆vz(m

/s)

D Control histories

Figure 6-3. Adaptive artificial potential function (AAPF) method results using CWHequation

The settling time obtained is 1715 s and the control effort at the settling time is

8.63 m/s. The running cost is shown in Figure 6-4, which illustrates how the cost

increases with every control action.

99

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

7

8

9

t (sec)

V(t)

Figure 6-4. AAPF running cost using CWH equation

The plot of the adaptive estimates is shown in Figure 6-5. These plots indicate that

the adaptive estimates continue to change depending on the chaser’s position in the

state space. This reinforces the fact that the magnitude of the commanded force should

depend on the dynamics and the location of the chaser.

0 1000 2000 30000

0.05

0.1

ρ11

0 1000 2000 3000−0.05

0

0.05

ρ12

0 1000 2000 3000−0.02

0

0.02

ρ13

0 1000 2000 3000−0.05

0

0.05

ρ22

0 1000 2000 3000−0.05

0

0.05

t (s)

ρ23

0 1000 2000 30000

2

4x 10

−3

t (s)

ρ33

Figure 6-5. AAPF adaptive estimates using CWH equation

6.3.2 Small Angle Approximation Example

The parameters used in this example are shown in Table 6-2. The goal of the

chaser (with initial relative orientation α0 to the target) is to reach a relative orientation

of α = 0 rad with the target that is tumbling at the rate ωt . The control constraint shown

100

below is enforced.

|τi | ≤ τmax for i = 1, 2, 3

Settling time is used as a performance metric where the trajectory obtained is

considered settled when ‖α‖ ≤ 0.01 rad. The running control effort cost is also used as

performance metric and is defined as

V (t) =

∫ tt0

‖τ‖1 dε .

Table 6-2. Relative orientation parametersParameter Value Units

J

300 20 1020 100 010 0 200

kg ·m2

ωt[0.25 −0.25 0.25

]T rad/sα0

[0.3 −0.2 −0.15

]T radα0

[0 0 0

]T rad/sαf

[0 0 0

]T radτmax 20 N ·mθ0 0.002×

[1 0 0 1 0 1

]T s−2

T 5 s

The results for this example are displayed in Figure 6-6. The relative orientation

and relative angular velocity plots in Figure 6-6A and Figure 6-6B, respectively, indicate

that both these parameters converge to zero. The corresponding control history is

given in Figure 6-6C. The settling time obtained is approximately 20.1 s. Note that the

running cost in Figure 6-6D increases linearly with time since the controller continues to

compensate for the tumbling satellite.

The plots of the adaptive estimates are shown in Figure 6-7. These plots indicate

that the adaptive estimates continue to change depending on the spacecrafts relative

orientation. This reinforces the fact that the magnitude of the commanded torque should

depend on the dynamics and the relative orientation.

101

0 20 40 60 80 100−0.5

0

0.5α1(rad)

0 20 40 60 80 100−0.4

−0.2

0

α2(rad)

0 20 40 60 80 100−0.2

0

0.2

t (s)

α3(rad)

A Position histories

0 20 40 60 80 100−0.1

0

0.1

α1(rad/s)

0 20 40 60 80 100−0.05

0

0.05

α2(rad/s)

0 20 40 60 80 100−0.05

0

0.05

t (s)

α3(rad/s)

B Velocity histories

0 20 40 60 80 100−20

−10

0

∆α1(rad/s)

0 20 40 60 80 1000

5

10

∆α

2(rad/s)

0 20 40 60 80 1000

10

20

t (s)

∆α

3(rad/s)

C Control histories

0 20 40 60 80 1000

500

1000

1500

2000

2500

3000

t (s)

V(t)

D Running cost

Figure 6-6. AAPF results using SAA equation

0 50 100−2

0

2

ρ11

0 50 100−2

0

2

ρ12

0 50 100−2

0

2

ρ13

0 50 100−2

0

2

ρ22

0 50 100−1

0

1

t (s)

ρ23

0 50 1000

0.5

1

t (s)

ρ33

Figure 6-7. AAPF adaptive estimates using SAA equation

102

In conclusion, a modification to the APF method is presented in this chapter.

This modification is done in an effort to implement a performance metric for tuning

the APF method. As a result, an adaptive update law is used to vary the weights on

the attractive APF. The velocity profile of the TPBVP is used in the derivation since

this solution is a good approximation to the fixed time minimum control effort problem.

A stability analysis is also presented, where it is demonstrated that convergence is

only guaranteed for obstacle-free (unconstrained) cases. Finally, two examples are

performed to demonstrate the AAPF method. The first is a relative translation problem in

a cluttered environment and the second is a relative orientation problem with a tumbling

spacecraft.

103

CHAPTER 7FINITE HORIZON LINEAR QUADRATIC PROBLEMS

In the previous chapter, the close-range rendezvous trajectories were addressed

by developing the adaptive artificial potential function (AAPF) method. However, the

AAPF method is not well-suited for the final approach or endgame of an autonomous

proximity operation (APO). Trajectories obtained from solving a linear quadratic (LQ)

optimal control problems (OCPs) are typically used for the final approach or endgame.

Moreover, solving a finite horizon LQ OCP augments the problem with a time constraint

which is necessary for certain APOs (i.e., intercept, rendezvous with uncooperative

target, docking, etc.). This chapter discusses the methods for solving finite horizon LQ

problems. First, the Differential Riccati Equation (DRE) is adopted from equation (4–15).

It is shown that there exists a system of linear matrix differential equations which

are intimately related to the DRE. Furthermore, when the Hamiltonian matrix is time

invariant, the DRE is solved using using a state transition matrix (STM) representation of

the linear system of matrix differential equations. When the Hamiltonian is time varying,

the solution cannot be obtained using a STM representation. As a result, two methods

of solving are developed using: Picard Iteration (PI) and Homotopy Continuation (HC).

A numerical example is presented to demonstrate the effectiveness of solving the

DRE for a LTV system using PI and HC. Finally, a final approach trajectory is obtained

using HC with the Yamanaka-Ankerson-Tschauner-Hempel (YATH) equation derived in

Appendix A.

7.1 Development

Trajectories for the final approach or endgame of an APO are typically obtained by

solving a LQ optimal control problem (OCP). This is because a linearized model serves

as a good approximation in this level of proximity and using a linearized model simplifies

the solution process. Moreover, the control law obtained from an LQ problem is in the

state feedback form which makes it compliant for implementation. It was shown in

104

Chapter 4 that solving a finite horizon LQ OCP in equation (4–10) amounts to solving a

final value problem with the DRE in equation (4–15). The difficulty here is that the DRE

is a nonlinear equation, which in general, cannot be analytically solved. Thus, solving a

final value problem with the DRE requires that the DRE be integrated backwards in time

every time a new solution is needed.

An interesting property of the DRE is that it is intimately related to the system of

linear matrix differential equationsU

V

=

A −BR−1BT

−Q −AT

U

V

,

U(tf )

V(tf )

=

I

Sf

, (7–1)

where V, U ∈ Rn×n. [91, 92] Note that the state matrix is the Hamiltonian matrix from

equation (4–12). If U−1 exists, then the solution to the DRE can be written using the

variable transformation

S = VU−1 . (7–2)

This is seen if equation (7–1) is substituted into the time derivative of equation (7–2) as

shown below.

S = VU−1 − VU−1UU−1

⇒ S =[−QU− ATV

]U−1 − VU−1

[AU− BR−1BTV

]U−1

⇒ S = −Q− AT(VU−1

)−(VU−1

)A−

(VU−1

)BR−1BT

(VU−1

)⇒ S = −Q− ATS− SA + SBR−1BTS

Note that the boundary condition on the DRE is also satisfied using the variable

transformation in equation (7–2) as shown below.

S(tf ) = V(tf )U−1(tf ) = Sf

105

Therefore, if the solution for U and V can be obtained, then the optimal control law

is defined using the variable transformation in equation (7–2). As a result, the optimal

control law from equation (4–16) is defined as

u = −R−1BTVU−1x .

7.2 Linear Time-Invariant System

When the Hamiltonian matrix is time invariant, equation (7–1) can be solved using a

STM representation. [92–94] In particular, for the constant Hamiltonian matrix

Hc =

A −BR−1BT

−Q −AT

,

the solution to equation (7–1) is written asU(t)

V(t)

= exp (Hc(t − tf ))

I

Sf

= Φ(t, tf )

I

Sf

, (7–3)

where Φ(t, tf ) is the STM of the system of linear matrix differential equations in

equation (7–1).

7.3 Linear Time-Varying System

When the Hamiltonian matrix is time varying, the STM is not easily solved for in

general. In principle, the STM does exist, however, the difficulty is in obtaining the actual

form of the STM. To address this difficulty, two methods are investigated for solving the

DRE: PI and HC. Both PI and HC are known to be convergent in general. [95, 96] Thus,

obtaining the solution using PI and/or HC can be used rather than having to integrate the

DRE backwards in time.

7.3.1 Picard Iteration

The PI is a fixed point iterative technique for solving differential or integral

equations. [95] Note that the final value problem with equation (7–1) is a differential

equation. Thus, if the Hamiltonian matrix is time varying, then the final value problem in

106

equation (7–1) can be solved using PI. First, the variable X is defined as

X =

U

V

,

so that equation (7–1) is written as

X = HtX , X(tf ) = Xf , (7–4)

where Ht is the time varying Hamiltonian matrix. Using PI, the solution to the (k + 1)th

iteration is written as

X(k+1)(t) = Xf +

∫ ttf

Ht(ε)X(k)(ε) dε .

The PI requires an initial guess (i.e., the fixed point). Using X(0) = 0 as the initial guess,

the solution as k →∞ is

X(t) =

[I +

∫ ttf

Ht(ε1)dε1 +

∫ ttf

Ht(ε2)

[∫ ε2

tf

Ht(ε1)dε1

]dε2 + ...

∫ ttf

Ht(ε3)

[∫ ε3

tf

Ht(ε2)

[∫ ε2

tf

Ht(ε1)dε1

]dε2

]dε3 + · · ·

]Xf

⇒ X(t) = Φ(t, tf )Xf .

The integral terms can be computed analytically using a symbolic manipulator

or a quadrature rule for a numerical approximation. However, analytic expressions

are favorable since the quadrature rule would have to be done at each time step and

recursively for each additional term. Note that if the Hamiltonian matrix is constant,

then the solution simply becomes the Taylor Series Expansion (TSE) of the matrix

exponential about t = tf (which is the solution in equation (7–3)). However, if the

Hamiltonian is not constant, then the Picard Iteration provides a solution for the STM.

107

7.3.2 Homotopy Continuation

Given a smooth nonlinear operator N(x), the HC can be used to solve

N(x) = 0 , (7–5)

which can not be analytically solved in general. [97] This is done by defining a

homotopy, which is a continuous topological map that injects L(x) → N(x) as the

embedding parameter p → 1. The idea is to continuously embed the solution of L(x) = 0

(which is known) to the solution of N(x) = 0. [96, 97] The benefit that HC has over other

perturbation techniques is that the radius and rate of convergence can be increased

depending on the parameters chosen in the solution formulation. [96] It is known that

HC is an effective tool for solving a wide range of problems (e.g., algebraic, ordinary

differential, partial differential, boundary value, etc.). [96] The approach for solving the

DRE using HC is discussed in this section.

First, the convex homotopy is defined

H(x, p) = (1− p) [L(x)− L(y0)]− phΨ(t) [N(x)] = 0 , (7–6)

where p ∈ ν ∈ R | 0 ≤ ν ≤ 1 is the embedding parameter, h 6= 0 is the auxiliary

parameter, Ψ(t) 6= 0 is the auxiliary function, and y0 is an initial guess. [96] Note that the

homotopy has the property

H(x, 0) = L(x)− L(y0) = 0

H(x, 1) = N(x) = 0 .

Next, the parameter x is written as a Maclaurin Series Expansion (MSE) in the

embedding parameter

x = x0 + px1 + p2x2 + · · · , (7–7)

108

where the parameters x0, x1, x2, ... are to be determined. Note that the solution to

N(x) = 0 can now be written as

x = limp→1

x0 + px1 + p2x2 + · · · =

∞∑i=0

xi .

Therefore, solving equation (7–5) amounts to solving for the parameters x0, x1, x2, ... .

These parameters are solved for by substituting equation (7–7) into equation (7–6).

Appendix C discusses the solution derivation using the original DRE as the operator

N(S) = S + Q− SBR−1BTS + ATS + SA = 0 , (7–8)

while choosing

L(S) = S . (7–9)

Appendix D discusses the solution derivation using the operator

N(X) = X−HtX = 0 , (7–10)

which is the system of linear matrix differential equations related to the DRE, and

choosing

L(X) = X . (7–11)

Appendix E discusses the solution derivation using equation (7–10) as the operator and

choosing

L(X) = X− X . (7–12)

Based on the formulation, there are free parameters that can be manipulated.

Namely, these are the operator L(x), the initial guess y0, the auxiliary parameter h, and

the auxiliary function Ψ(t). These parameters can be chosen such that the radius and

rate of convergence is improved (depending on the problem being solved). [96] In the

109

solution derivations in the appendices, these parameters are chosen by default (except

for the auxiliary parameter h). However, a thorough analysis should be performed to

determine the set of parameters that yield the desired results.

A feature of the PI and HC is the parametric solution structure of the Riccati

matrix. Since the solutions are functions of the time horizon and the state, input, and

LQ matrices, then the solution is easily updated. This makes the solution structure

amenable to changes in the time horizon, dynamics model, and/or performance index.

7.4 Numerical Examples

An example is presented to demonstrate how well the solution to the DRE can be

approximated using PI and the HC. This example is used since linearized models of

dynamical systems tend to have oscillatory components in their dynamics. In particular,

this is true for models of relative translational motion in orbit (as shown in Appendix A).

[98, 99]

A two degrees-of-freedom linear system is used with the dynamicsx1x2

=

0 1

sin(t) 0

x1x2

+

0

1

u ,

x1(t0)x2(t0)

=

2

1

⇒ x = A(t)x + Bu , x(t0) = x0

The parameters and matrices that are associated with the finite horizon LQ problem are

given in Table 7-1.

Table 7-1. Finite horizon linear quadratic parametersParameters Value Units

Sf I ∈ R2×2 –Q I ∈ R2×2 –R I ∈ R1×1 –t0 0 stf 20 s

110

7.4.1 Picard Iteration

Using the PI approach to solve the DRE, the results are shown in Figure 7-1. The

results were computed for 0th to 7th order approximations. The four components of

the Riccati matrix S are shown in Figure 7-1A thru Figure 7-1D, where S∗i ,j indicates

the optimal solution. The plots indicate that adding an order of approximation does

not necessarily yield a better result. In fact, it is shown that only the even ordered

approximations yielded favorable estimates. The max difference between the components

of S(k) and S∗ is shown in Table 7-2. It is shown that increasing the order of approximation

did improve the solution about the boundary condition at t = tf .

Table 7-2. Maximum difference of Riccati matrix components (PI)Approximation order ∆S1,1 ∆S1,2 ∆S2,2 ∆S2,2

0 2.28 2.29 2.29 1.351 46.04 44.36 45.66 43.822 2.21 2.10 2.05 1.193 22.99 14.88 20.52 13.074 2.11 1.84 1.89 1.035 6.35e3 4.44e3 6.09e3 4.26e36 2.12 1.80 1.73 1.047 90.45 68.80 80.38 61.00

The resulting solutions were then used to determine the control, and in turn

the response of the states. The states are shown in Figure 7-1E and the control is

shown in Figure 7-1F, where x∗ and u∗ indicate the optimal states and optimal control,

respectively. The states converge to the optimal solution, but at the cost of a large

initial control value. The set of states that converged faster were for the even numbered

approximations (since the Riccati matrices for these cases were fair approximations).

The resulting costs obtained for each approximation are given in Table 7-3 where J∗

indicates the optimal cost. The lowest cost was obtained for the 3rd order approximation,

however, this is still far from the optimal cost. Note that the 5th order approximation

resulted in a large cost. It was determined that this was caused by the U becoming

ill-conditioned.

111

0 5 10 15 20

0

5

10

15S1,1

t (s)

S ∗1,1

S(0)1,1

S(1)1,1

S(2)1,1

S(3)1,1

S(4)1,1

S(5)1,1

S(6)1,1

S(7)1,1

A S11

0 5 10 15 20

0

5

10

15

S1,2

t (s)

S ∗1,2

S(0)1,2

S(1)1,2

S(2)1,2

S(3)1,2

S(4)1,2

S(5)1,2

S(6)1,2

S(7)1,2

B S12

0 5 10 15 20

0

5

10

15

S2,1

t (s)

S ∗2,1

S(0)2,1

S(1)2,1

S(2)2,1

S(3)2,1

S(4)2,1

S(5)2,1

S(6)2,1

S(7)2,1

C S21

0 5 10 15 20

0

5

10

15

S2,2

t (s)

S ∗2,2

S(0)2,2

S(1)2,2

S(2)2,2

S(3)2,2

S(4)2,2

S(5)2,2

S(6)2,2

S(7)2,2

D S22

0 5 10 15 20−5

0

5

10

x1

x∗

x(0)

x(1)

x(2)

x(3)

x(4)

x(5)

x(6)

x(7)

0 5 10 15 20−5

0

5

x2

t (S)

E States

0 5 10 15 20−10

−8

−6

−4

−2

0

2

4

6

8

10

u

t (S)

u∗

u(0)

u(1)

u(2)

u(3)

u(4)

u(5)

u(6)

u(7)

F Control

Figure 7-1. Picard Iteration results

112

Table 7-3. Costs for each approximating methodCost PI HC (case 1) HC (case 2) HC (case 3)J∗ 8.44 8.44 8.44 8.44

J(0) 223.18 223.18 223.18 223.18

J(1) 92.34 229.20 31.93 15.67

J(2) 91.35 240.02 41.18 11.13

J(3) 29.78 243.15 8.80 11.70

J(4) 59.12 250.03 11.73 11.73

J(5) 1.25e32 282.32 9.36 11.41

J(6) 43.76 369.90 8.90 -.-J(7) 57.43 -.- -.- -.-

7.4.2 Homotopy Continuation

Three different cases of HC were attempted for solving the DRE. The first case is

for the homotopy defined by the operators in equation (7–8) and equation (7–9). The

results obtained are shown in Figure 7-2, where the value for the auxiliary parameter

used is h = −1/20. The results were computed for the 0th to 6th order approximations.

The four components of the Riccati matrix S are shown in Figure 7-2A thru Figure 7-2D.

The results indicate that this HC approach does not improve as the order of the

approximation increases. Also, the results obtained do not seem to have the same

trend as the optimal solution. The max difference between the components of S(k) and

S∗ is shown in Table 7-4.

Table 7-4. Maximum difference of Riccati matrix components (HC case 1)Approximation order ∆S1,1 ∆S1,2 ∆S2,2 ∆S2,2

0 2.28 2.29 2.29 1.351 1.97 1.67 2.29 1.352 1.68 2.50 2.28 1.343 1.92 3.39 2.26 1.334 2.99 4.09 2.24 1.325 4.08 4.68 2.22 1.366 5.16 5.21 2.22 1.51

The resulting solutions are then used to determine the control, and in turn the

response of the states. The states are shown in Figure 7-2E and the control is shown

in Figure 7-2F. The results converge for each case, but do not necessarily improve

113

as the order of approximation increases. This can be improved by choosing a better

combination of the operator L(S), the auxiliary parameter h, the auxiliary function

Ψ(t), and the initial approximation Y0. The resulting costs obtained for each order

approximation are given in Table 7-3. The best results were obtained for the 1st order

approximation since this gave the best approximation for the Riccati matrix at the

initial time. Notice that since the DRE is being solved directly, there is no concern for

ill-conditioned matrices.

The second case is for the homotopy defined by the operators in equation (7–10)

and equation (7–11). The results obtained are shown in Figure 7-3 where the value

for the auxiliary parameter used is h = −1/2. The results were computed for the 0th

to 6th order approximations. The four components of the Riccati matrix S are shown in

Figure 7-3A thru Figure 7-3D. The results indicate that this HC approach does have

consistent improvement as the order of the approximation increases. The results tend

to have the same trend as the optimal solution as well. The max difference between the

components of S(k) and S∗ is shown in Table 7-5.

Table 7-5. Maximum difference of Riccati matrix components (HC case 2)Approximation order ∆S1,1 ∆S1,2 ∆S2,2 ∆S2,2

0 2.28 2.29 2.29 1.351 12.92 12.17 12.54 11.672 2.13 1.88 1.84 0.983 1.03 0.99 0.99 0.484 1.55 1.16 1.13 0.395 1.12 1.00 0.92 0.466 1.05 0.84 0.72 0.26

The resulting solutions were then used to determine the control, and in turn the

response of the states. The states are shown in Figure 7-3E and the control is shown in

Figure 7-3F. The results do converge and stay near the optimal solution as the order of

the approximation increases. The resulting costs obtained for each order approximation

are given in Table 7-3. The best results were obtained for the 3rd order approximation.

This again might be coincidental since this approximation might have yielded a good

114

0 5 10 15 20−1

0

1

2

3

4

5

6S1,1

t

S ∗1,1

S(0)1,1

S(1)1,1

S(2)1,1

S(3)1,1

S(4)1,1

S(5)1,1

S(6)1,1

A S11

0 5 10 15 20−1

0

1

2

3

4

5

6

S1,2

t

S ∗1,2

S(0)1,2

S(1)1,2

S(2)1,2

S(3)1,2

S(4)1,2

S(5)1,2

S(6)1,2

B S12

0 5 10 15 20−1

0

1

2

3

4

5

6

S2,1

t

S ∗2,1

S(0)2,1

S(1)2,1

S(2)2,1

S(3)2,1

S(4)2,1

S(5)2,1

S(6)2,1

C S21

0 5 10 15 20−1

0

1

2

3

4

5

6

S2,2

t

S ∗2,2

S(0)2,2

S(1)2,2

S(2)2,2

S(3)2,2

S(4)2,2

S(5)2,2

S(6)2,2

D S22

0 5 10 15 20−5

0

5

10

15

x1

x∗

x(0)

x(1)

x(2)

x(3)

x(4)

x(5)

x(6)

0 5 10 15 20−10

−5

0

5

10

x2

t (s)

E States

0 5 10 15 20−6

−4

−2

0

2

4

6

8

10

u

t

u∗

u(0)

u(1)

u(2)

u(3)

u(4)

u(5)

u(6)

F Control

Figure 7-2. Homotopy continuation (case 1) results (h = −1/20)

115

approximation for the initial value of the Riccati matrix. Despite this, it is shown that the

homotopy used yields a good approximation to the solution to the DRE.

The third case is for the homotopy defined by the operators in equation (7–10)

and equation (7–12). The results obtained are shown in Figure 7-4 where the value

for the auxiliary parameter used is h = −1/2. The results were computed for the 0th

to 5th order approximations. The four components of the Riccati matrix S are shown

in Figure 7-4A thru Figure 7-4D. The results indicate that this HC solution shows

improvement immediately after the 0th approximation. However, the improvements with

the subsequent approximations seem to plateau. However, the results do have the same

trend as the optimal solution. The max differences between the components of S(k) and

S∗ are shown in Table 7-6.

Table 7-6. Maximum difference of Riccati matrix components (HC case 3)Approximation order ∆S1,1 ∆S1,2 ∆S2,2 ∆S2,2

0 2.28 2.29 2.29 1.351 1.53 1.54 1.54 1.012 1.34 1.16 1.14 0.633 1.44 1.18 1.16 0.554 1.44 1.19 1.16 0.565 1.41 1.17 1.14 0.56

The resulting solutions were then used to determine the control, and in turn the

response of the states. The states are shown in Figure 7-4E and the control is shown in

Figure 7-4F. The results do converge and stay near the optimal solution as the order of

the approximation increases. The resulting costs obtained for each order approximation

are given in Table 7-3. The best results were obtained for the 2nd order approximation.

This again might be coincidental since this approximation might have yielded a good

approximation for the initial value of the Riccati matrix.

7.5 Yamanaka-Ankerson-Tschauner-Hempel Example

An example using the YATH equation derived in Appendix A is presented in this

section. Based on the example in the previous section, the HC (Case 2) yielded the

116

0 5 10 15 20−1

0

1

2

3

4

5

6

7

8

9

10S1,1

t

S ∗1,1

S(0)1,1

S(1)1,1

S(2)1,1

S(3)1,1

S(4)1,1

S(5)1,1

S(6)1,1

A S11

0 5 10 15 20−1

0

1

2

3

4

5

6

7

8

9

10

S1,2

t

S ∗1,2

S(0)1,2

S(1)1,2

S(2)1,2

S(3)1,2

S(4)1,2

S(5)1,2

S(6)1,2

B S12

0 5 10 15 20−1

0

1

2

3

4

5

6

7

8

9

10

S2,1

t

S ∗2,1

S(0)2,1

S(1)2,1

S(2)2,1

S(3)2,1

S(4)2,1

S(5)2,1

S(6)2,1

C S21

0 5 10 15 20−1

0

1

2

3

4

5

6

7

8

9

10

S2,2

t

S ∗2,2

S(0)2,2

S(1)2,2

S(2)2,2

S(3)2,2

S(4)2,2

S(5)2,2

S(6)2,2

D S22

0 5 10 15 20−5

0

5

10

x1

x∗

x(0)

x(1)

x(2)

x(3)

x(4)

x(5)

x(6)

0 5 10 15 20−5

0

5

x2

t (s)

E States

0 5 10 15 20−6

−4

−2

0

2

4

6

u

t

u∗

u(0)

u(1)

u(2)

u(3)

u(4)

u(5)

u(6)

F Control

Figure 7-3. Homotopy continuation (case 2) results (h = −1/2)

117

0 5 10 15 20−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4S

1,1

t

S ∗1,1

S(0)1,1

S(1)1,1

S(2)1,1

S(3)1,1

S(4)1,1

S(5)1,1

A S11

0 5 10 15 20−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

S1,2

t

S ∗1,2

S(0)1,2

S(1)1,2

S(2)1,2

S(3)1,2

S(4)1,2

S(5)1,2

B S12

0 5 10 15 20−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

S2,1

t

S ∗2,1

S(0)2,1

S(1)2,1

S(2)2,1

S(3)2,1

S(4)2,1

S(5)2,1

C S21

0 5 10 15 20−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

S2,2

t

S ∗2,2

S(0)2,2

S(1)2,2

S(2)2,2

S(3)2,2

S(4)2,2

S(5)2,2

D S22

0 5 10 15 20−5

0

5

10

x1

t (s)

x∗

x(0)

x(1)

x(2)

x(3)

x(4)

x(5)

0 5 10 15 20−5

0

5

x2

t (s)

E States

0 5 10 15 20−6

−4

−2

0

2

4

6

u

t (s)

u∗

u(0)

u(1)

u(2)

u(3)

u(4)

u(5)

F Control

Figure 7-4. Homotopy continuation (case 3) results (h = −1/2)

118

closest result to the optimal solution. In this section, the HC (Case 2) approach shown in

Appendix D is used where the solution is obtained up to the fourth order approximation.

The value of the auxiliary parameter used is h = −1/10. The parameters associated

with the LQ problem are given in Table 7-7. Note that the independent variable for this

set of equations is the true anomaly of the target.

Table 7-7. Final approach parametersParameters Value Units

Sf

[103 × I 0

0 10−3 × I

]∈ R6×6 –

Q

[103 × I 0

0 10−2 × I

]∈ R6×6 –

R 10−3 × I ∈ R3×3 –e 0.15 –k 0.0256

√rad/s

r0[10 −10 10

]T mv0

[0 −0 0

]T m/sθ0 0 degθf 10 deg

The results obtained are shown in Figure 7-5. The relative position history is shown

in Figure 7-5A where the black line represents the optimal solution and the other lines

represent the different approximation orders. This figure shows that the solutions begin

to converge after the first order approximation. In addition, the solution gets close to

the optimal solution as the order of the approximation increases. The relative velocity

histories are shown in Figure 7-5B. These results indicate that the velocity converges

to zero after the first order approximation as well. The corresponding control histories

are shown in Figure 7-5C and a zoomed view of this plot is shown in Figure 7-5D. These

plots show that the approximations require a relatively large initial control action. This is

likely due to a poor approximation to the Riccati matrix at the initial time. Despite this,

the trajectories were shown to converge to the origin. The costs associated with each

trajectory are shown in Table 7-8. This table shows the cost continues to decrease as

the order of the approximation increases.

119

0 50 100 150 200

0

5

10

x(m

)

0 50 100 150 200

−10

−5

0

y(m

)

r∗

r(0)

r(1)

r(2)

r(3)

r(4)

0 50 100 150 200

0

5

10

z(m

)

t (s)

A Position histories

0 50 100 150 200

−0.4

−0.2

0

vx(m

/s)

0 50 100 150 200

0

0.2

0.4

vy(m

/s)

v∗

v (0)

v (1)

v (2)

v (3)

v (4)

0 50 100 150 200

−0.4

−0.2

0

vz(m

/s)

t (s)

B Velocity histories

0 50 100 150 200

−0.4

−0.2

0

ux(m

/s2)

0 50 100 150 200

0

0.2

0.4

uy(m

/s2)

u∗

u(0)

u(1)

u(2)

u(3)

u(4)

0 50 100 150 200

−0.4

−0.2

0

uz(m

/s2)

t (s)

C Control histories

0 50 100 150 200−0.01

0

0.01ux(m

/s2)

0 50 100 150 200−0.01

0

0.01

uy(m

/s2)

u∗

u(0)

u(1)

u(2)

u(3)

u(4)

0 50 100 150 200−0.01

0

0.01

uz(m

/s2)

t (s)

D Control histories (zoomed view)

Figure 7-5. Homotopy continuation results for final approach example (h = −1/10)

Table 7-8. Costs for final approach using HCCost HC (case 2)J∗ 6.74e3

J(0) 1.78e5

J(1) 7.72e4

J(2) 7.41e4

J(3) 7.13e4

J(4) 7.08e4

120

In this chapter, the methods for solving a finite horizon LQ problem for both an LTI

and LTV systems are discussed. For the case where the Hamiltonian matrix is constant,

it is shown that solution is determined by obtaining the STM of linear system of matrix

differential equations related to the DRE. When the Hamiltonian matrix is not constant,

two methods are presented for obtaining the solution. For the numerical example shown

in this chapter, the best result is obtained by using the HC approach with the mapping

between equation (7–10) and equation (7–11). This mapping is then used to solve

the final approach problem with the YATH equations presented in Appendix A. It is

shown that the after the first order approximation, the solutions converge to the origin. In

addition, as the order of the approximation increased, the solution matches more closely

to the optimal solution.

121

CHAPTER 8NUMERICAL RESULTS

The results from the numerical analyses performed are presented and discussed

in this chapter. First, the results of the Monte Carlo simulations (MCSs) performed

characterize the performance and convergence characteristics for the Artificial Potential

Function (APF) and Adaptive Artificial Potential Function (AAPF) methods. The

MCSs are performed with both the Clohessy-Wiltshire-Hill (CWH) and small angle

approximation (SAA) equations. Next, the results of implementing the optimal solutions

obtained in Chapter 4 in the high fidelity model are presented. Two methods of tracking

are used: “trajectory tracking” and “force tracking”. The aim of this study is to determine

how well the spacecraft is able to track the trajectories obtained and to characterize

the discrepancies between the optimal trajectory and the realized trajectory. The APF

and AAPF methods are also implemented in the high fidelity model to determine how

these methods perform in a higher fidelity model. Next, a close-range rendezvous

scenario with obstacle avoidance is presented. The scenario is simulated with a planned

trajectory, the APF method, and the AAPF method. Lastly, a final approach scenario is

presented which uses the finite horizon LQ solution using the CWH equation and the

state transition matrix (STM) solution presented in Chapter 7.

8.1 Monte Carlo Simulations

It was shown in Chapter 6 that a stability proof for the APF and AAPF methods only

exists for the obstacle-free case. Moreover, there is no methodology to characterize

the performance of these methods. As a result, MCSs are performed to demonstrate

performance and convergence characteristics of the APF and AAPF methods. Two

MCSs are performed for relative translation (i.e., CWH equation) and relative orientation

(i.e., SAA equation).

In the relative translation MCS, the initial conditions of the chaser are varied. The

initial conditions are chosen as normally distributed random vectors with r0 ∼ N (0, 500I),

122

v0 ∼ N (0, I), and with the constraints

250 m ≤ ‖r0‖ ≤ 1000 m

‖r0 − ri‖ ≥ 50 m for i = 1, 2, 3, ... , 14 .

These constraints are enforced to ensure that there is a considerable separation

initially between the chaser, target, and obstacles while maintaining the close proximity

assumption. A set of 14 static obstacles are placed in the state space at the positions

shown below.

r1 =

[500 0 0

]Tr2 =

[−500 0 0

]Tr3 =

[0 500 0

]Tr4 =

[0 −500 0

]Tr5 =

[0 0 500

]Tr6 =

[0 0 −500

]Tr7 =

[250 250 250

]Tr8 =

[250 −250 250

]Tr9 =

[250 250 −250

]Tr10 =

[250 −250 −250

]Tr11 =

[−250 250 250

]Tr12 =

[−250 −250 250

]Tr13 =

[−250 250 −250

]Tr14 =

[−250 −250 −250

]TThe initial positions and obstacles are illustrated in Figure 8-1A where the blue circles

represent the initial positions and the black dots represent the static obstacles. The

parameters used in this MCS are given in Table 8-1.

Table 8-1. Monte Carlo simulation parameters (CWH equation)Parameter APF Value AAPF Value Units

n 0.0012 0.0012 rad/srf 0 0 mfmax 1 1 m/sP I ∈ R3×3 – s−2

θ0 – 0.002×[1 0 0 1 0 1

]T s−2

Ni I ∈ R3×3 I ∈ R3×3 s−2

ψi 1.5× 105 1.5× 105 –σi 1.0× 104 1.0× 104 m2

k 0.002 0.002 –T – 1000 s

123

In the relative orientation MCS, the initial conditions of the chaser and tumbling

rate of the target are varied. The initial conditions were chosen as normally distributed

random vectors with α0 ∼ N (0, 0.1I), α0 ∼ N (0, 0.1I), ωt ∼ N (0, 0.05I), and the

constraints

0.1 rad ≤ ‖α0‖ ≤ 0.25 rad

‖α0‖ ≤ 0.25 rad/s

‖ωt‖ ≤ 0.25 rad/s .

The constraints are enforced to maintain the small angle assumption yet have a distinct

initial relative orientation and tumbling rate between the chaser and target. Obstacles

are not included in this MCS. However, they could have been included to represent

“keep out zones” in the relative orientation states. The initial orientations used are

illustrated in Figure 8-1B. The parameters used in this MCS are given in Table 8-2.

Table 8-2. Monte Carlo simulation parameters (SAA equation)Parameter APF Value AAPF Value Units

αf 0 0 radτmax 20 20 m/s

P I ∈ R3×3 – s−2

θ0 – 0.002×[1 0 0 1 0 1

]T s−2

k 2 – –T – 5 s

Settling time is used to characterize convergence where the solution is assumed to

have settled when r ≤ 10 m for the relative translation case and α ≤ 0.01 rad for the

relative orientation case. Performance is characterized based on the control effort cost

J =

∫ tft0

‖u‖1dt ,

where u is the control parameter and tf is the settling time.

The results obtained for the relative translation MCS are shown in Figure 8-2. The

cost data is shown in Figure 8-2A where the average cost obtained using the APF

124

−1000

−500

0

500

1000

−1000

−500

0

500

1000

−1000

−500

0

500

1000

x (m)y (m)

z(m

)

A CWH initial positions and obstacle positions

−0.2−0.1

00.1

0.2

−0.2

0

0.2

−0.2

−0.1

0

0.1

0.2

α1 (rad)α2 (rad)

α3(rad)

B SAA initial orientations

Figure 8-1. Initial relative position/orientations used in Monte Carlo simulations

method is 8.60 m/s and the average cost obtained using the AAPF method is 4.12 m/s.

The settle time data is shown in Figure 8-2B where the average settling time obtained

with the APF method is 1315 s and the average settling time obtained using the AAPF

method was 1209 s. These results indicate that the AAPF method commands less

control effort than the APF method in the relative translation scenario. In addition, the

AAPF method is able to specify a time constraint based on the largely skewed (to the

right) data in Figure 8-2B for the AAPF method. A large majority of the data points

centered around the 1000 s is a result of choosing the transfer time T = 1000 s. The

results of the MCS indicate that the AAPF method commands less control effort globally

while being able to choose a time criterion in the transfer time T .

The results obtained for the relative orientation MCS are shown in Figure 8-2. The

cost data is shown in Figure 8-2C where the average cost obtained using the APF

method is 207.7 N · m while the average cost obtained using the AAPF method is

65.03 N ·m. The settle time data is shown in Figure 8-2D where the average settling time

obtained using the APF method is 6.85 s and the average settling time obtained using

the AAPF method is 8.97 s. These results indicate that the AAPF method commands

less control effort than the APF method in the relative orientation scenario. However, a

125

firm time constraint is not seen based on the transfer time chosen for the AAPF method.

This is due to the linearized model (i.e., SAA equation) and the constant compensation

for the tumbling rates of each sample point. Although the average settling time was

slightly larger for the AAPF method, the control effort was considerably lower.

0 5 10 15 200

10

20

30

40

APF Cost

Frequen

cy

0 2 4 6 8 100

10

20

30

40

AAPF Cost

Frequen

cy

A CWH cost data

500 1000 1500 2000 2500 30000

10

20

30

APF Settle Time (s)

Frequen

cy

500 1000 1500 2000 2500 30000

200

400

600

800

AAPF Settle Time (s)

Frequen

cy

B CWH settling time data

0 200 400 600 800 1000 12000

20

40

60

APF Cost

Frequen

cy

0 50 100 150 200 250 300 350 4000

20

40

60

80

AAPF Cost

Frequen

cy

C SAA cost data

0 10 20 30 40 500

50

100

APF Settle Time (s)

Frequen

cy

0 5 10 15 20 25 30 35 400

50

100

AAPF Settle Time (s)

Frequen

cy

D SAA settle time data

Figure 8-2. Costs and settling times data from Monte Carlo simulations

8.2 High Fidelity Simulations

The simulation results presented in this section are generated using the high

fidelity model discussed in Chapter 3. The parameters used for the target and chaser

spacecraft are given in Table 8-3 and Table 8-4, respectively. In the simulations, the

optimal trajectories obtained in Chapter 4 are implemented using two methods of

126

tracking: “trajectory tracking” and “force tracking”. Since the trajectories obtained in

Chapter 4 are only defined at the collocation points, interpolation is used for both

tracking methods to obtain a continuous reference signal.

Table 8-3. Target spacecraft parametersParameter Value Unitsmt 100 kglt 3 m

rt(t0)[−1182.959348 6817.396210 904.495486

]T kmvt(t0)

[0.175776 −0.963776 7.494102

]T km/s

Jt

300 20 1020 100 010 0 200

kg ·m2

qt(t0)[0 0 0 1

]T –ωt(t0)

[0 0 0

]T rad/s

Table 8-4. Chaser spacecraft parametersParameter Value Unitsmc 100 kglc 3 m

rc(t0)[−1182.339411 6816.939420 904.891745

]T kmvc(t0)

[0.175776 −0.963776 7.494102

]T km/sfmax

√3 m/s2

fdb 0.01 m/s2

Jc

300 20 1020 100 010 0 200

kg ·m2

qc(t0)[1 0 0 0

]T –ωc(t0)

[0 0 0

]T rad/sΩmax 5000 rad/sΩmax 100 rad/s2

The “trajectory tracking” method involves tracking the optimal states obtained from

the optimal solution. A cubic spline is used to interpolate between the collocation points

of the optimal trajectory since cubic splines are smooth (i.e., ensure the derivative of

the interpolating polynomial is continuous) and do not have the oscillatory behavior of

127

high-degree polynomial interpolation. [73] To track a position, the error is defined as

ed = r∗c − rc ,

where r∗c is the optimal position of the chaser. As a result, the proportional-integral-derivative

(PID) controller

f = kped + ki

∫ tt0

ed dε + kd ed

is used, where kp = 0.6, ki = 0.06, and kd = 1.5 are the tuned PID controller gains. [71]

The PID controller defines the force required to track the optimal chaser position based

on the true chaser position. To track an orientation, the error quaternion is defined

according to equation (2–8)

qe = q∗c ⊗ q−1c ,

where q∗c is the optimal attitude trajectory of the chaser. As a result, the eigenaxis

attitude controller

τ = −kJcεe − cJcωc + ω×c Jcωc

is used, where εe is the vector component of the error quaternion and k = 9.54, c = 5.5

are gains. [100] The eigenaxis controller generates the torque required to perform an

eigenaxis slew (i.e., “minimum eigenangle” slew) towards the optimal chaser orientation

based on the true chaser orientation. [100]

The “force tracking” method involves tracking the optimal control and torque profiles

using the respective actuators. A piecewise cubic Hermite interpolating polynomial

is used to interpolate between the collocation points of the control. This method of

interpolation is chosen since the optimal control is sometimes discontinuous and

piecewise polynomials can handle this phenomenon. [73] As a result, the interpolated

optimal control is fed forward to the actuators as the commanded control action.

128

The APF and AAPF methods are also implemented in the high fidelity model.

These simulations are performed to determine how well the APF and AAPF methods

perform in a high fidelity model. Both methods are designed to “re-plan” trajectories

until the terminal conditions are met. A close-range rendezvous scenario with obstacle

avoidance is also simulated. The trajectory obtained in Chapter 4 is implemented using

“trajectory tracking”. The APF and AAPF methods are also implemented to demonstrate

close-range rendezvous with obstacle avoidance.

The last set of results presented are the final approach scenario. The final approach

is done using the finite horizon LQ solution using the CWH equation. The feedback

control law for this approach is developed in Chapter 7. This controller is implemented to

demonstrate how using a linearized model can be used to develop a feedback controller

which is implementable in higher fidelity dynamics.

8.2.1 Minimum Time Trajectories

Figure 8-3 shows the results of implementing the minimum time trajectories using

“trajectory tracking” in the high fidelity model. The magnitude of the relative position

is shown in Figure 8-3A where the blue line represents the relative position between

the chaser and target and the green line represents the relative position between the

chaser and the optimal trajectory. This figure indicates that the optimal trajectory is

tracked within 5 m and the distance between the chaser and the optimal trajectory is

approximately 0.5 m at the final time. However, the distance between the chaser and the

target is approximately 10 m. This deviation is small and is due to disturbances and/or

inaccuracies in the optimal solution.

The orientation error is shown in Figure 8-3B where the blue line represents the

error angle between the chaser and target and the green line represents the error

angle between the chaser and optimal trajectory. This figure indicates that the attitude

trajectory can be tracked within a 10 error angle. The error angle between the chaser

129

and target and between the chaser and optimal trajectory is virtually the same (i.e.,

about 1) at the final time. The deviation is due inaccuracies in the optimal solution.

The relative position plots are shown in Figure 8-3C and the quaternion trajectories

are shown in Figure 8-3D. Both match closely to the optimal trajectories, which indicates

that the respective dynamic models used are accurate for minimum time trajectories.

The force plot is shown in Figure 8-3E and the torque plot is shown in Figure 8-3F.

The force plot indicates that “trajectory tracking” leads to a control history similar to

the optimal control history. The torque plot indicates that “trajectory tracking” does not

necessarily lead to a control history similar to the optimal control history. In particular,

the first component of the torque matches the optimal control; however, the second and

third components do not.

Next, Figure 8-4 shows the results of implementing the minimum time trajectories

using “force tracking” with the high fidelity model. The relative position magnitude is

shown in Figure 8-4A where the blue line represents the relative position between the

chaser and target and the green line represents the relative position between the chaser

and the optimal trajectory. This figure indicates that the optimal force does not yield the

same response in a higher fidelity model. At the final time, the relative position between

the chaser and the optimal trajectory is approximately 10 m and the relative position

between the chaser and the target is approximately 3 m. It is equivocal as to why “force

tracking” yields a better response in the high fidelity model.

The orientation error plot is shown in Figure 8-4B where the blue line represents

the error angle between the chaser and target and the green line represents the error

angle between the chaser and the optimal trajectory. This figure indicates that the error

angle between the chaser and target and between the chaser and the optimal trajectory

are virtually the same (approximately 5) at the final time. This is evidence that the

optimal control does not yield the same response in the high fidelity model and as a

result the terminal conditions are not satisfied. This is likely due to a drift in the attitude

130

0 10 20 30 4010

−3

10−2

10−1

100

101

102

103

‖rrel‖

(m)

t (s)

‖rrel‖‖r ∗

rel‖

A Position tracking error

0 10 20 30 4010

−6

10−4

10−2

100

102

104

θe(d

eg)

t (s)

θeθ∗e

B Angle tracking error

0 10 20 30 40−1000

0

1000

rrel,x(m

)

0 10 20 30 40−1000

0

1000

rrel,y(m

)

rrelr∗rel

0 10 20 30 40−1000

0

1000

rrel,z(m

)

t (s)

C Relative position

0 10 20 30 40

−1

0

1

ǫ 1

0 10 20 30 40

−1

0

1

ǫ 2

qc

q∗

0 10 20 30 40

−1

0

1

ǫ 3

0 10 20 30 40

−1

0

1

ηe

t (s)

D Error quaternion

0 10 20 30 40

−1

0

1

fx(m

/s2)

0 10 20 30 40

−1

0

1

fy(m

/s2)

ff ∗

0 10 20 30 40

−1

0

1

fz(m

/s2)

t (s)

E Force history

0 10 20 30 40

−2

0

2

τx(N

m)

0 10 20 30 40

−2

0

2

τy(N

m)

ττ ∗

0 10 20 30 40

−2

0

2

τz(N

m)

t (s)

F Torque history

Figure 8-3. Minimum time “trajectory tracking” results

131

of the target caused by disturbances. Another source of error might be the “bang-bang”

control structure. This control structure requires that a discontinuous jump be made in

the control which is not physically realizable with the reaction wheels.

The relative position plots are shown in Figure 8-4C and the quaternion trajectories

are shown in Figure 8-4D. Both plots indicate that the resultant trajectory is similar

to the optimal trajectory. This indicates that the models used for generating the

optimal solutions are accurate for minimum time problems. The force plot is shown

in Figure 8-4E and the torque plot is shown in Figure 8-4F. These plots indicate that the

optimal control can be tracked accurately by the actuators.

8.2.2 Fixed Time Minimum Control Effort Trajectories

Figure 8-5 shows the results of implementing the fixed time minimum control effort

trajectories using “trajectory tracking” with the high fidelity model. The magnitude of

the relative position is shown in Figure 8-5A where the blue line represents the relative

position between the chaser and target and the green line represents the relative

position between the chaser and the optimal trajectory. This figure indicates that the

optimal trajectory is tracked accurately (i.e., within a 1 m error). The plot shows a high

frequency low amplitude tracking signal which implies the PID controller is accurately

tracking the trajectory. However, the position of the chaser is not nearly the same as the

position of the target (approximately 1 km away) at the final time. This is due to secular

disturbing effects which were not modeled in the reference trajectories.

The orientation error is shown in Figure 8-5B where the blue line represents the

error angle between the chaser and target and the green line represents the error angle

between the chaser and optimal trajectory. This error plot is evidence that the optimal

attitude trajectory can be tracked within a 1 error angle and that the final orientation

matches the results from the higher fidelity model (within a 0.1 error angle) at the final

time.

132

0 10 20 30 4010

−3

10−2

10−1

100

101

102

103

‖rrel‖

(m)

t (s)

‖rrel‖‖r ∗

rel‖

A Position tracking error

0 10 20 30 4010

−6

10−4

10−2

100

102

104

θe(d

eg)

t (s)

θeθ∗e

B Angle tracking error

0 10 20 30 40−1000

0

1000

rrel,x(m

)

0 10 20 30 40−1000

0

1000

rrel,y(m

)

rrelr∗rel

0 10 20 30 40−1000

0

1000

rrel,z(m

)

t (s)

C Relative position

0 10 20 30 40

−1

0

1

ǫ 1

0 10 20 30 40

−1

0

1

ǫ 2

qc

q∗

0 10 20 30 40

−1

0

1

ǫ 3

0 10 20 30 40

−1

0

1

ηe

t (s)

D Error quaternion

0 10 20 30 40

−1

0

1

fx(m

/s2)

0 10 20 30 40

−1

0

1

fy(m

/s2)

ff ∗

0 10 20 30 40

−1

0

1

fz(m

/s2)

t (s)

E Force history

0 10 20 30 40−2

0

2

τ x(N

m)

0 10 20 30 40−2

0

2

τ y(N

m)

ττ ∗

0 10 20 30 40−2

0

2

τ z(N

m)

t (s)

F Torque history

Figure 8-4. Minimum time “force tracking” results

133

The relative position plots shown in Figure 8-5C illustrate the secular disturbing

effects. The trajectories are initially coincident and begin to drift after approximately

100 s. Since the disturbances affect the chaser and the target, the error goes unaccounted

for in both spacecraft. This result is motivation for a need to “re-plan” for “unmodeled”

effects.

The quaternion trajectories shown in Figure 8-5D match closely which agrees

with the results in Figure 8-5B. This indicates that the model used for rotational motion

is accurate. The force plot is shown in Figure 8-5E and the torque plot is shown in

Figure 8-5F. Both the force and torque plots indicate that “trajectory tracking” leads to a

control history similar to the optimal control history.

Next, Figure 8-6 shows the results of implementing the fixed time minimum control

effort trajectories using “force tracking” with the high fidelity model. The relative position

magnitude is shown in Figure 8-6A where the blue line represents the relative position

between the chaser and target and the green line represents the relative position

between the chaser and the optimal trajectory. The green line indicates that optimal

control does not yield the same response in a higher fidelity model. This is due to the

secular disturbing effects acting on the chaser. However, the chaser is closer to the

target at the final time (approximately 300 m). This is a result of the “bang-off-bang”

control structure being fully captured when using “force tracking”.

The orientation error plot is shown in Figure 8-6B where the blue line represents

the error angle between the chaser and target and the green line represents the error

angle between the chaser and the optimal trajectory. This figure indicates that the

error between the chaser and target and the error between the chaser and the optimal

trajectory are about the same (approximately 10) at the final time. This is evidence

that the optimal control history does not yield the desired terminal conditions in a

higher fidelity model. This may be due to a drift in the orientation of the target or the

“bang-off-bang” control structure. Since there is a finite width in between the switching

134

0 100 200 300 400 50010

−6

10−4

10−2

100

102

104

‖rrel‖

(m)

t (s)

‖rrel‖‖r ∗

rel‖

A Position tracking error

0 20 40 60 80 10010

−6

10−4

10−2

100

102

104

θe(d

eg)

t (s)

θeθ∗e

B Angle tracking error

0 100 200 300 400 500−1000

0

1000

rrel,x(m

)

0 100 200 300 400 500−1000

0

1000

rrel,y(m

)

rrelr∗rel

0 100 200 300 400 500−1000

0

1000

rrel,z(m

)

t (s)

C Relative position

0 20 40 60 80 100

−1

0

1

ǫ 1

0 20 40 60 80 100

−1

0

1

ǫ 2

qc

q∗

0 20 40 60 80 100

−1

0

1

ǫ 3

0 20 40 60 80 100

−1

0

1

ηe

t (s)

D Error quaternion

0 100 200 300 400 500

−0.5

0

0.5

fx(m

/s2)

0 100 200 300 400 500

−0.5

0

0.5

fy(m

/s2)

ff ∗

0 100 200 300 400 500

−0.5

0

0.5

fz(m

/s2)

t (s)

E Force history

0 20 40 60 80 100

−1

0

1

τ x(N

m)

0 20 40 60 80 100

−0.2

0

0.2

0.4

τ y(N

m)

ττ ∗

0 20 40 60 80 100−0.2

0

0.2

0.4

τ z(N

m)

t (s)

F Torque history

Figure 8-5. Fixed time minimum control effort “trajectory tracking” results

135

points of the control, the interpolated values at the switching points add an error to the

response.

The relative position plots shown in Figure 8-6C illustrate the secular disturbing

effects. It is shown that the trajectories are initially coincident and begin to drift after

approximately 100 s. This figure also illustrates that the terminal condition obtained

using “force tracking” are better than those obtained using “trajectory tracking”. However,

the error at the final time is still large. The quaternion plot shown in Figure 8-6D

indicates that the the trajectories are similar, yet there is a 10 error at the terminal

condition which is evident in Figure 8-6B. The force plot is shown in Figure 8-6E and

the torque plot is shown in Figure 8-6F. These plots indicate that the optimal control

histories are tracked accurately by the actuators.

8.2.3 Finite Horizon Quadratic Cost Trajectories

Figure 8-7 shows the results of implementing the finite horizon quadratic cost

trajectories using “trajectory tracking” with the high fidelity model. The magnitude of

the relative position is shown in Figure 8-7A where the blue line represents the relative

position between the chaser and target and the green line represents the relative

position between the chaser and the optimal trajectory. This figure indicates that the

optimal trajectory is tracked accurately (i.e., under a 0.01 m error). However, the relative

position between the chaser and target is not close to zero (approximately 1 km) at the

final time. This is again due to secular disturbing effects.

The orientation error is shown in Figure 8-7B where the blue line represents the

error angle between the chaser and target and the green line represents the error

angle between the chaser and optimal trajectory. This plot is evidence that the optimal

trajectory can be tracked within a 1 error angle, and that the final orientation matches

the results from the higher fidelity model (within a 0.1 error) at the final time. This result

indicates that disturbances have less effect on the rotational motion.

136

0 100 200 300 400 50010

−3

10−2

10−1

100

101

102

103

104

‖rrel‖

(m)

t (s)

‖rrel‖‖r ∗

rel‖

A Position tracking error

0 20 40 60 80 10010

−6

10−4

10−2

100

102

104

θe(d

eg)

t (s)

θeθ∗e

B Angle tracking error

0 100 200 300 400 500−1000

0

1000

rrel,x(m

)

0 100 200 300 400 500−1000

0

1000

rrel,y(m

)

rrelr∗rel

0 100 200 300 400 500−1000

0

1000

rrel,z(m

)

t (s)

C Relative position

0 20 40 60 80 100

−1

0

1

ǫ 1

0 20 40 60 80 100

−1

0

1

ǫ 2

qc

q∗

0 20 40 60 80 100

−1

0

1

ǫ 3

0 20 40 60 80 100

−1

0

1

ηe

t (s)

D Error quaternion

0 100 200 300 400 500

−0.5

0

0.5

fx(m

/s2)

0 100 200 300 400 500

−0.5

0

0.5

fy(m

/s2)

ff ∗

0 100 200 300 400 500

−0.5

0

0.5

fz(m

/s2)

t (s)

E Force history

0 20 40 60 80 100

−1

0

1

τ x(N

m)

0 20 40 60 80 100

−0.2

0

0.2

0.4

τ y(N

m)

ττ ∗

0 20 40 60 80 100−0.2

0

0.2

0.4

τ z(N

m)

t (s)

F Torque history

Figure 8-6. Fixed time minimum control effort “force tracking” results

137

The relative position plots shown in Figure 8-7C illustrate how the chaser drifts as

it tracks the optimal trajectory. Initially the trajectories are coincident and begin to drift

after approximately 100 s. The quaternion trajectories shown in Figure 8-7D match

closely, which is reinforced by the results in Figure 8-7B. The force plot is shown in

Figure 8-7E and the torque plot is shown in Figure 8-7F. The force plot indicates that

“trajectory tracking” leads to a control history similar to the optimal control history. The

torque plot is another example where “trajectory tracking” does not necessarily lead to a

control history similar to the optimal control history.

Next, Figure 8-8 shows the results of implementing the finite horizon quadratic

cost trajectories using “force tracking” with the high fidelity model. The relative position

magnitude is shown in Figure 8-8A where the blue line represents the relative position

between the chaser and target and the green line represents the relative position

between the chaser and the optimal trajectory. The green line indicates that optimal

control does not yield the same response in a higher fidelity model. However, the chaser

is closer to the target at the final time (approximately 300 m). This is again due to the

secular disturbing effects.

The orientation error plot is shown in Figure 8-8B where the blue line represents

the error angle between the chaser and target and the green line represents the error

angle between the chaser and the optimal trajectory. This figure indicates that the

error between the chaser and target and the error between the chaser and the optimal

trajectory are about the same (about 10) at the final time. This is evidence that the

optimal control history does not yield the terminal conditions in a higher fidelity model.

This is due to a drift in the orientation of the target and the initial “bang-bang” control

structure when the control is saturated.

The relative position plots shown in Figure 8-8C illustrate the secular disturbing

effects. The trajectories are initially coincident and begin to drift after approximately

100 s. The quaternion plots shown in Figure 8-8D indicate that the trajectories

138

0 100 200 300 400 50010

−6

10−4

10−2

100

102

104

‖rrel‖

(m)

t (s)

‖rrel‖‖r ∗

rel‖

A Position tracking error

0 20 40 60 80 10010

−6

10−4

10−2

100

102

104

θe(d

eg)

t (s)

θeθ∗e

B Angle tracking error

0 100 200 300 400 500−1000

0

1000

rrel,x(m

)

0 100 200 300 400 500−1000

0

1000

rrel,y(m

)

rrelr∗rel

0 100 200 300 400 500−1000

0

1000

rrel,z(m

)

t (s)

C Relative position

0 20 40 60 80 100

−1

0

1

ǫ 1

0 20 40 60 80 100

−1

0

1

ǫ 2

qc

q∗

0 20 40 60 80 100

−1

0

1

ǫ 3

0 20 40 60 80 100

−1

0

1

ηe

t (s)

D Error quaternion

0 100 200 300 400 500

−0.2

0

0.2

fx(m

/s2)

0 100 200 300 400 500

−0.2

0

0.2

fy(m

/s2)

ff ∗

0 100 200 300 400 500

−0.2

0

0.2

fz(m

/s2)

t (s)

E Force history

0 20 40 60 80 100−2

0

2

τ x(N

m)

0 20 40 60 80 100−2

0

2

τ y(N

m)

ττ ∗

0 20 40 60 80 100−2

0

2

τz(N

m)

t (s)

F Torque history

Figure 8-7. Finite horizon quadratic cost “trajectory tracking” results

139

are similar. The force plot is shown in Figure 8-8E and the torque plot is shown in

Figure 8-8F. These plots indicate that the optimal control histories can be tracked

accurately by the actuators.

8.2.4 Disturbance-Free Trajectories

To determine the extent to which disturbances, inaccuracies in optimal solutions,

and the control structure contribute to the drifting trajectories, two additional examples

are performed while disabling disturbances. The first example is implementing the fixed

time minimum control effort trajectories using “trajectory tracking”. Figure 8-9 shows

the results obtained for this example. The magnitude of the relative position is shown in

Figure 8-9A where the blue line represents the relative position between the chaser and

target and the green line represents the relative position between the chaser and the

optimal trajectory. This figure indicates that the optimal trajectory is tracked accurately

(i.e., under a 0.1 m error). The relative position between the chaser and target and

between the chaser and optimal trajectory are virtually the same (approximately 0.3 m)

at the final time. Figure 8-9C also supports this since both trajectories are similar and

they both are near the origin at the final time.

The orientation error is shown in Figure 8-7B where the blue line represents the

error angle between the chaser and target and the green line represents the error

angle between the chaser and optimal trajectory. The quaternion plots are shown in

Figure 8-9D. The effects of removing disturbances is not evident in these plots since

disturbances did not have a large effect on the rotational motion in the first place. The

force plot is shown in Figure 8-9E and the torque plot is shown in Figure 8-9F. The

control history here is similar to the control history with disturbances.

The next example is implementing the finite horizon quadratic cost trajectories

using “force tracking”. Figure 8-10 shows the results for this example. The results

indicate that the “force tracking” method does not give favorable results without

disturbances. Figure 8-10A and Figure 8-10C show that the terminal conditions are

140

0 100 200 300 400 50010

−3

10−2

10−1

100

101

102

103

104

‖rrel‖

(m)

t (s)

‖rrel‖‖r ∗

rel‖

A Position tracking error

0 20 40 60 80 10010

−6

10−4

10−2

100

102

104

θe(d

eg)

t (s)

θeθ∗e

B Angle tracking error

0 100 200 300 400 500−1000

0

1000

rrel,x(m

)

0 100 200 300 400 500−1000

0

1000

rrel,y(m

)

rrelr∗rel

0 100 200 300 400 500−1000

0

1000

rrel,z(m

)

t (s)

C Relative position

0 20 40 60 80 100

−1

0

1

ǫ 1

0 20 40 60 80 100

−1

0

1

ǫ 2

qc

q∗

0 20 40 60 80 100

−1

0

1

ǫ 3

0 20 40 60 80 100

−1

0

1

ηe

t (s)

D Error quaternion

0 100 200 300 400 500

−0.2

0

0.2

fx(m

/s2)

0 100 200 300 400 500

−0.2

0

0.2

fy(m

/s2)

ff ∗

0 100 200 300 400 500

−0.2

0

0.2

fz(m

/s2)

t (s)

E Force history

0 20 40 60 80 100−2

0

2

τ x(N

m)

0 20 40 60 80 100−2

0

2

τ y(N

m)

ττ ∗

0 20 40 60 80 100−2

0

2

τz(N

m)

t (s)

F Torque history

Figure 8-8. Finite horizon quadratic cost “force tracking” results

141

0 100 200 300 400 50010

−3

10−2

10−1

100

101

102

103

‖rrel‖

(m)

t (s)

‖rrel‖‖r ∗

rel‖

A Position tracking error

0 20 40 60 80 10010

−6

10−4

10−2

100

102

104

θe(d

eg)

t (s)

θeθ∗e

B Angle tracking error

0 100 200 300 400 500−1000

0

1000

rrel,x(m

)

0 100 200 300 400 500−1000

0

1000

rrel,y(m

)

rrelr∗rel

0 100 200 300 400 500−1000

0

1000

rrel,z(m

)

t (s)

C Relative position

0 20 40 60 80 100

−1

0

1

ǫ 1

0 20 40 60 80 100

−1

0

1

ǫ 2

qc

q∗

0 20 40 60 80 100

−1

0

1

ǫ 3

0 20 40 60 80 100

−1

0

1

ηe

t (s)

D Error quaternion

0 100 200 300 400 500

−0.5

0

0.5

fx(m

/s2)

0 100 200 300 400 500

−0.5

0

0.5

fy(m

/s2)

ff ∗

0 100 200 300 400 500

−0.5

0

0.5

fz(m

/s2)

t (s)

E Force history

0 20 40 60 80 100

−1

0

1

τ x(N

m)

0 20 40 60 80 100

−0.2

0

0.2

0.4

τ y(N

m)

ττ ∗

0 20 40 60 80 100−0.2

0

0.2

0.4

τ z(N

m)

t (s)

F Torque history

Figure 8-9. Fixed time minimum control effort “trajectory tracking” results (nodisturbances)

142

not obtained. Moreover, Figure 8-10B and Figure 8-10D show results similar to the

results in Figure 8-8 which include disturbances. Thus, deviations using “force tracking”

are caused by inaccuracies in the optimal solutions. These inaccuracies are due to the

interpolation since the solutions are only known at the collocation points. This result is

motivation for the need to “re-plan” when there are inaccuracies in a solution.

8.2.5 Artificial Potential Function Trajectory

The target and chaser parameters used are the same as given in Table 8-3

and Table 8-4, respectively. The APF parameters used for this example are give in

Table 8-5. Figure 8-11 shows the results of implementing the APF method in the high

fidelity model. Figure 8-11A is a plot of the magnitude of the relative position versus

time. This figure indicates that the magnitude of the relative position continues to

decrease with every maneuver. Figure 8-11B shows the relative position history and the

corresponding control history is shown in Figure 8-11C. These results indicate that the

chaser continuously approaches the target with every maneuver.

Table 8-5. APF method parametersParameter Value Units

rf 0 mP I ∈ R3×3 s−2

k 0.002 –

8.2.6 Adaptive Artificial Potential Function Trajectory

The target and chaser parameters used are the same as given in Table 8-3 and

Table 8-4, respectively. The AAPF parameters used for this example are give in

Table 8-6. Figure 8-12 shows the results of implementing the AAPF method in the

high fidelity model. Figure 8-12A is a plot of the magnitude of the relative position

versus time. This figure indicates that the magnitude of the relative position continues

to decrease and that a maneuver is executed approximately every 200 s. This is no

coincidence since the transfer time chosen was T = 200 s. Figure 8-12B shows the

relative position history and the corresponding control history is shown in Figure 8-12C.

143

0 100 200 300 400 50010

−3

10−2

10−1

100

101

102

103

‖rrel‖

(m)

t (s)

‖rrel‖‖r ∗

rel‖

A Position tracking error

0 20 40 60 80 10010

−6

10−4

10−2

100

102

104

θe(d

eg)

t (s)

θeθ∗e

B Angle tracking error

0 100 200 300 400 500−1000

0

1000

rrel,x(m

)

0 100 200 300 400 500−1000

0

1000

rrel,y(m

)

rrelr∗rel

0 100 200 300 400 500−1000

0

1000

rrel,z(m

)

t (s)

C Relative position

0 20 40 60 80 100

−1

0

1

ǫ 1

0 20 40 60 80 100

−1

0

1

ǫ 2

qc

q∗

0 20 40 60 80 100

−1

0

1

ǫ 3

0 20 40 60 80 100

−1

0

1

ηe

t (s)

D Error quaternion

0 100 200 300 400 500

−0.2

0

0.2

fx(m

/s2)

0 100 200 300 400 500

−0.2

0

0.2

fy(m

/s2)

ff ∗

0 100 200 300 400 500

−0.2

0

0.2

fz(m

/s2)

t (s)

E Force history

0 20 40 60 80 100−2

0

2

τ x(N

m)

0 20 40 60 80 100−2

0

2

τ y(N

m)

ττ ∗

0 20 40 60 80 100−2

0

2

τz(N

m)

t (s)

F Torque history

Figure 8-10. Finite horizon quadratic cost “force tracking” results (no disturbances)

144

0 200 400 600 800 100010

−1

100

101

102

103

t (s)

‖rrel‖

(m)

A Position tracking error

0 200 400 600 800 1000

−500

0

500

rrel,x(m

)

0 200 400 600 800 1000

−500

0

500

rrel,y(m

)

0 200 400 600 800 1000

−500

0

500

t (s)

rrel,z(m

)

B Relative position

0 200 400 600 800 1000−2

0

2

fx(m

/s2)

0 200 400 600 800 1000−2

0

2

fy(m

/s2)

0 200 400 600 800 1000−2

0

2

t (s)

fz(m

/s2)

C Control history

Figure 8-11. APF method results

These results indicate that the chaser continuously approaches the target with every

maneuver. Moreover, the AAPF method is suitable for rendezvous in a high fidelity

model despite its development using a linearized model.

Table 8-6. AAPF method parametersParameter Value Units

rf 0 mθ0 0.002×

[1 0 0 1 0 1

]T s−2

T 200 s

145

0 200 400 600 800 100010

0

101

102

103

t (s)

‖rrel‖

(m)

A Position tracking error

0 200 400 600 800 1000

−500

0

500

rrel,x(m

)

0 200 400 600 800 1000

−500

0

500

rrel,y(m

)

0 200 400 600 800 1000

−500

0

500

t (s)

rrel,z(m

)

B Relative position

0 200 400 600 800 1000−2

0

2

fx(m

/s2)

0 200 400 600 800 1000−2

0

2

fy(m

/s2)

0 200 400 600 800 1000−2

0

2

t (s)

fz(m

/s2)

C Control history

Figure 8-12. AAPF method results

8.2.7 Obstacle Avoidance Trajectories

The results obtained for the close-range rendezvous scenario with obstacle

avoidance are presented in this section. The chaser and target have the same

parameters shown in Table 8-4 and Table 8-3, respectively. The obstacle has the

parameters shown in Table 8-7 and is modeled the same as the chaser and target. The

optimal solution obtained in Chapter 4 is implemented using “trajectory tracking”.

The results obtained from implementing the constrained fixed time minimum control

effort trajectory using “trajectory tracking” are shown in Figure 8-13 and Figure 8-14.

The relative position trajectories of the chaser are shown at different instances in

146

Table 8-7. Obstacle spacecraft parametersParameter Value Unitsmo 100 kglo 3 m

ro(t0)[−1182.649497 6817.167778 904.693252

]T kmvo(t0)

[0.175775 −0.963776 7.494102

]T km/s

Jo

300 20 1020 100 010 0 200

kg ·m2

qo(t0)[0 0 0 1

]T –ωo(t0)

[0 0 0

]T rad/s

Figure 8-13A thru Figure 8-13D. These figures support the results that were shown

previously. That is, while the optimal trajectory can tracked accurately, there is disparity

between the optimal trajectory and the actual trajectory. In this case, tracking the

trajectory did not result in a collision between the chaser and the target. However, if the

chaser did approach the obstacle due to drifting effects, then a correction would must be

made to the trajectory to avoid a collision.

The magnitude of the relative position is shown in Figure 8-14A where the blue

line represents the relative position between the chaser and the target and the green

line represents the relative position between the chaser and the optimal trajectory.

This figure indicates that the trajectory can be tracked accurately. The relative position

plots shown in Figure 8-14B illustrate the drifting that occurs from tracking a planned

trajectory. The force plot is shown in Figure 8-14C. The commanded force is similar to

the optimal force solution which is the same result for the previous “trajectory tracking”

results presented.

The parameters used for implementing the APF method for close-range rendezvous

with obstacle avoidance are shown in Table 8-8. Recall that the path constraint for

the obstacle required that the chaser be more that 100 m from the obstacle at all

times. Thus, the APF parameters are chosen such that the repulsive potential roughly

represents a “sphere” with radius of 100 m.

147

0

500

1000

−500−400

−300−200

−1000

0

100

200

300

400

500

rrel,y (m)

Time: 125 sec

rrel,x (m)

rrel,z(m

)

A t = 125 s

0

500

1000

−500−400

−300−200

−1000

0

100

200

300

400

500

rrel,y (m)

Time: 250 sec

rrel,x (m)

rrel,z(m

)

B t = 250 s

0

500

1000

−500−400

−300−200

−1000

0

100

200

300

400

500

rrel,y (m)

Time: 375 sec

rrel,x (m)

rrel,z(m

)

C t = 375 s

0

500

1000

−500−400

−300−200

−1000

0

100

200

300

400

500

rrel,y (m)

Time: 500 sec

rrel,x (m)

rrel,z(m

)

D t = 500 s

Figure 8-13. Constrained fixed time minimum control effort trajectories

148

0 100 200 300 400 50010

−6

10−4

10−2

100

102

104

‖rrel‖

(m)

t (s)

‖rrel‖‖r ∗

rel‖

A Position tracking error

0 100 200 300 400 500−1000

0

1000

rrel,x(m

)

0 100 200 300 400 500−1000

0

1000

rrel,y(m

)

rrelr∗rel

0 100 200 300 400 500−1000

0

1000

rrel,z(m

)

t (s)

B Relative position

0 100 200 300 400 500

−0.5

0

0.5

fx(m

/s2)

0 100 200 300 400 500

−0.5

0

0.5

fy(m

/s2)

ff ∗

0 100 200 300 400 500

−0.5

0

0.5

fz(m

/s2)

t (s)

C Force history

Figure 8-14. Constrained fixed time minimum control effort “trajectory tracking” results

Table 8-8. Constrained APF method parametersParameter Value Units

rf 0 mP I ∈ R3×3 s−2

N I ∈ R3×3 s−2

ψ 2.5× 106 –σ 2.5× 103 m2

k 0.004 –

149

The results obtained from implementing the APF method are shown in Figure 8-15

and Figure 8-16. The relative position trajectories of the chaser are shown at different

instances in Figure 8-15A thru Figure 8-15D. These figures indicate that the chaser is

able to rendezvous with the target while avoiding the obstacle. Figure 8-15A shows that

the chaser encounters the obstacle after approximately 250 s and performs a maneuver

to avoid it.

The magnitude of the relative position between the chaser and the target is shown

in Figure 8-16A. This figure shows that the chaser continues to approach the target

and gets within 8 m of the target after 1000 s. The relative position plots are shown in

Figure 8-16B and the corresponding control histories are shown in Figure 8-16C. These

force plot shows the avoidance maneuver that occurred at approximately 150 s. As a

result, the relative position plot shows that this avoidance maneuver keeps the chaser

from approaching the target until the next maneuver.

The parameters used for implementing the AAPF method for close-range rendezvous

with obstacle avoidance are shown in Table 8-9. The same parameters for the repulsive

potential are used for this simulation. The gain k is smaller since it is only applied to the

gradient of the repulsive potential and the scaling between the attractive and repulsive

potentials is different than the APF method.

Table 8-9. Constrained AAPF method parametersParameter Value Units

rf 0 mθ0 0.002×

[1 0 0 1 0 1

]T s−2

T 200 sN I ∈ R3×3 s−2

ψ 2.5× 106 –σ 2.5× 103 m2

k 0.001 –

The results obtained from implementing the AAPF method are shown in Figure 8-17

and Figure 8-18. The relative position trajectories of the chaser are shown at different

instances in Figure 8-17A thru Figure 8-17D. These figures show that the chaser is able

150

0

500

1000

400200

0

0

100

200

300

400

500

rrel,y (m)

Time: 250 sec

rrel,x (m)

rrel,

z(m

)

A t = 250 s

0

500

1000

400200

0

0

100

200

300

400

500

rrel,y (m)

Time: 500 sec

rrel,x (m)

rrel,

z(m

)

B t = 500 s

0

500

1000

−400−200

0

0

100

200

300

400

500

rrel,y (m)

Time: 750 sec

rrel,x (m)

rrel,z(m

)

C t = 750 s

0

500

1000

−400−200

0

0

100

200

300

400

500

rrel,y (m)

Time: 1000 sec

rrel,x (m)

rrel,z(m

)

D t = 1000 s

Figure 8-15. Constrained APF method trajectories

151

0 200 400 600 800 100010

0

101

102

103

t (s)

‖rrel‖

(m)

A Position tracking error

0 200 400 600 800 1000

−500

0

500

rrel,x(m

)

0 200 400 600 800 1000

−500

0

500

rrel,y(m

)

0 200 400 600 800 1000

−500

0

500

t (s)

rrel,z(m

)

B Relative position

0 200 400 600 800 1000−2

0

2

fx(m

/s2)

0 200 400 600 800 1000−2

0

2

fy(m

/s2)

0 200 400 600 800 1000−2

0

2

t (s)

fz(m

/s2)

C Force history

Figure 8-16. Constrained APF method results

to rendezvous with the target while avoiding the obstacle. In this case, the obstacle is

sensed sooner than the APF method. This is due to the different scaling between the

attractive and repulsive potentials. Smaller values of ψ and σ can be used to reduce the

effect of the repulsive potentials and to allow the chaser to converge to the target at a

faster rate.

The magnitude of the relative position between the chaser and target is shown in

Figure 8-18A. This figure shows that the chaser approaches the target at a faster rate

than the APF method. This is expected since it was shown that convergence times are

improved with the AAPF method. The relative position plots are shown in Figure 8-18B

152

0

500

1000

−400−200

0

0

100

200

300

400

500

rrel,y (m)

Time: 250 sec

rrel,x (m)

rrel,z(m

)

A t = 250 s

0

500

1000

−400−200

0

0

100

200

300

400

500

rrel,y (m)

Time: 500 sec

rrel,x (m)

rrel,z(m

)

B t = 500 s

0

500

1000

−400−200

0

0

100

200

300

400

500

rrel,y (m)

Time: 750 sec

rrel,x (m)

rrel,z(m

)

C t = 750 s

0

500

1000

−400−200

0

0

100

200

300

400

500

rrel,y (m)

Time: 1000 sec

rrel,x (m)

rrel,z(m

)

D t = 1000 s

Figure 8-17. Constrained AAPF method trajectories

153

and the corresponding control histories are shown in Figure 8-18C. The force plot shows

that the avoidance maneuver occurs at approximately 100 s. As a result, the relative

position plot shows that the obstacle is avoided yet the chaser continues to approach the

target.

0 200 400 600 800 100010

0

101

102

103

t (s)

‖rrel‖

(m)

A Position tracking error

0 200 400 600 800 1000

−500

0

500

rrel,x(m

)

0 200 400 600 800 1000

−500

0

500

rrel,y(m

)

0 200 400 600 800 1000

−500

0

500

t (s)

rrel,z(m

)

B Relative position

0 200 400 600 800 1000−2

0

2

fx(m

/s2)

0 200 400 600 800 1000−2

0

2

fy(m

/s2)

0 200 400 600 800 1000−2

0

2

t (s)

fz(m

/s2)

C Force history

Figure 8-18. Constrained AAPF method results

8.2.8 Final Approach Trajectory

The results obtained for the final approach scenario are presented in this section.

The final approach is performed using the finite horizon LQ control law. The CWH

equation from Appendix A is used in the LQ problem and the Riccati matrix is obtained

using the STM representation. The parameters used are given in Table 8-10 which

154

include the parameters defining the Riccati matrix and a new set of initial positions and

velocities for the chaser and target.

Table 8-10. Final approach parametersParameter Value Units

rt(t0)[−1005.448303 5813.480084 3725.077734

]T kmvt(t0)

[0.697715 −3.976259 6.387082

]T km/srt(t0)

[−1005.453976 5813.510411 3725.067550

]T kmvt(t0)

[0.697738 −3.976498 6.386375

]T km/s

Sf

[10−1 × I 0

0 10−1 × I

]∈ R6×6 –

Q

[10−1 × I 0

0 10−1 × I

]∈ R6×6 –

R 102 × I ∈ R3×3 –tf 50 s

The results of implementing the finite horizon LQ feedback control law are shown in

Figure 8-19. The magnitude of the relative position between the chaser and the target

is shown in Figure 8-19A. This figure shows that the chaser is able to perform the final

approach with the target and gets within 0.2 m. The relative position and relative velocity

plots are shown in Figure 8-19B and Figure 8-19C, respectively. These plots show that

using a linearized model to develop a control law for the final approach is valid. The

corresponding control histories are shown in Figure 8-19D. The effects of the deadband

constraints on the actuator are seen in this figure when the control values are near zero.

This is also seen in Figure 8-19A since the deadband limits do not allow the relative

position to decrease below 0.2 m.

In conclusion, the MCSs presented in this chapter suggest that the AAPF method

developed in Chapter 6 is more easily tuned than the APF method. A single scalar

parameter is required to tune the AAPF method where this parameter balances the

transfer time versus the control effort. Also, faster convergence times with less control

effort are seen since the dynamics are embedded in the formulation of the AAPF

method.

155

0 20 40 60 80 10010

−1

100

101

102

t (s)

‖rrel‖

(m)

A Position tracking error

0 20 40 60 80 100

−4

−2

0

rrel,x(m

)

0 20 40 60 80 100

0

10

20

30

rrel,y(m

)

0 20 40 60 80 100

−10

−5

0

t (s)

rrel,z(m

)

B Relative position

0 20 40 60 80 1000

0.2

0.4

vrel,x(m

/s)

0 20 40 60 80 100

−2

−1

0

vrel,y(m

/s)

0 20 40 60 80 100

−0.5

0

0.5

t (s)

vrel,z(m

/s)

C Relative velocity

0 20 40 60 80 100

0

0.05

0.1

0.15

fx(m

/s2)

0 20 40 60 80 100

−0.8

−0.6

−0.4

−0.2

0

fy(m

/s2)

0 20 40 60 80 100

0

0.2

0.4

t (s)

fz(m

/s2)

D Force history

Figure 8-19. Final approach results

The high fidelity simulation results suggest that using implementing planned

trajectories is only suitable for a certain classes of problems. The rotational motion in

the close-range rendezvous scenario yielded favorable results for the three pertinent

problems (at most a 10 error). The translational motion only yielded favorable results

for the minimum time problems since the transfer time was short. The cases with a long

transfer time (e.g., minimum control effort and finite horizon quadratic cost examples)

experienced secular disturbing effects which greatly affected the performance and

terminal conditions. Moreover, inaccuracies are introduced since the optimal solution

is only known at the collocation points. An interpolation scheme is used to obtain a

156

continuous solution; however, the interpolated points are not necessarily part of the

optimal solution.

The APF and AAPF methods are also implemented in the high fidelity model. The

results indicate that these methods can be used for close-range rendezvous scenarios

(with and without obstacles). In particular, the AAPF method is able to converge faster

than the APF method despite it being developed using a linearized model. A final

approach scenario is also simulated using the finite horizon LQ control law. This method

also performed favorably despite the control law being developed using a linearized

model.

157

CHAPTER 9CONCLUSIONS

The technology for enabling path-planning on board space systems for autonomous

proximity operations (APOs) is developed in this manuscript. Proximity operations

are bifurcated into two phases: (i) close-range rendezvous and (ii) final approach

or endgame. The motivation for developing algorithms that compute trajectories in

real-time is developed since this must be to done to account for “unmodeled” events. As

a result, autonomous space systems must have the capability of planning or correcting

trajectories on-demand while optimizing the solutions (with respect to a performance

index).

The adaptive artificial potential function (AAPF) method was developed for the

close-range rendezvous scenario. This method is a modification to the artificial potential

function (APF) method which embeds the dynamics and a performance criterion in the

formulation. This modification was done since the APF method is not well-suited for

a conservative system (with respect to resources) such as a spacecraft. The AAPF

method was shown to have improved performance and convergence times over the APF

method through Monte Carlo simulations (MCSs).

It is also shown through extensive numerical analyses that tracking planned

trajectories is ineffective for APOs when the transfer times of the trajectories are long.

This implies that using short time horizons to plan trajectories must be done to account

for all disturbances or “unmodeled” events. Moreover, the short time horizon framework

lends itself well to using the AAPF method since it is developed using a linearized model

which is accurate for short time horizons. This was demonstrated in simulation which

proves that the AAPF method can be used with a high fidelity model.

Final approach trajectories are conventionally obtained by solving a finite horizon

linear quadratic (LQ) problem which essentially is solved as a final value problem

with a Differential Riccati equation (DRE). It is shown that if the Hamiltonian matrix is

158

linear time invariant (LTI), then the solution is obtained using a state transition matrix.

However, this can not be done (in general) when the Hamiltonian matrix is linear time

varying (LTV). As a result, two mathematical tools were utilized to avoid solving a final

value problem and in turn obtain the solution to the DRE in real-time. Namely, these

tools are the Picard Iteration (PI) and the Homotopy Continuation (HC). The parametric

solution structure obtained using the PI and the HC allow for updates to be easily made

in the time horizon, dynamics model, and/or performance index. A numerical example

was presented to demonstrate how solutions are obtained using the PI and HC. In

addition, an example of a final approach scenario in an elliptic orbit (i.e., using the

Yamanaka-Ankerson-Tschauner-Hempel equation) was presented with the HC method.

Lastly, a final approach trajectory is implemented in the high fidelity model using an LQ

control law which demonstrated how using a linear model is valid for this scenario.

In conclusion, the methods developed in this manuscript are well-suited for

autonomous space systems. The methods developed compute optimized trajectories

through functional evaluations rather than through iterative techniques. To further

increase the range of applicability, different models can be used in the development (like

those described in [98, 99, 101]) which are amenable with the methods developed. An

instructional effect from developing these methodologies is that path-planning can be

done as the properties of the dynamics become more well-established.

159

APPENDIX ARELATIVE MOTION MODELS

The models to describe relative motion of two spacecraft are discussed in this

appendix. The translational relative motion model is derived using the two body model

on both the chaser and target. Using the close proximity assumption, the model results

in a linear form. Two sets of equations are presented to describe relative translational

motion when the target’s orbit is: (i) circular and (ii) eccentric. The rotational relative

motion model is derived using a small angle approximation (SAA) which also results in a

linear form.

A.1 Relative Translation

The position of the chaser relative to the target is

r = rc − rt ,

where rc is the position of the chaser and rt is the position of the target. If both

spacecraft are governed by the two body relative motion model

rt = − µ⊕‖rt‖3

rt

rc = − µ⊕‖rc‖3

rc + f ,

and the chaser is the only spacecraft with control thrust f, then the relative position is

differentiated twice to yield

r = rc − rt

⇒ r = − µ⊕‖rc‖3

rc + f +µ⊕‖rt‖3

rt

⇒ r = − µ⊕‖r + rt‖3

(r + rt) + f +µ⊕‖rt‖3

rt . (A–1)

160

Note that the value ‖r + rt‖ can be simplified as

‖r + rt‖ =[(r + rt)

T (r + rt)] 12 =

[‖rt‖2 + 2rT rt + ‖r‖2

] 12

⇒ ‖r + rt‖3 = ‖rt‖3[

1 + 2rT rt‖rt‖2

+‖r‖2‖rt‖2

] 32

.

This identity is substituted into equation (A–1) to yield

r = − µ⊕‖rt‖3

[1 + 2

rT rt‖rt‖2

+‖r‖2‖rt‖2

]− 32

(r + rt) + f +µ⊕‖rt‖3

rt . (A–2)

Moreover, the polynomial expansion

(1 + x)k = 1 + kx +k(k − 1)

2!x2 + · · ·

is used to expand[1 +

(2

rT rt‖rt‖2

+‖r‖2‖rt‖2

)]− 32

= 1− 3

2

(2

rT rt‖rt‖2

+‖r‖2‖rt‖2

)+ · · · .

Using the close proximity assumption (i.e., rt r) and the polynomial expansion above,

equation (A–2) is simplified as

r = − µ⊕‖rt‖3

(r + rt − 3

rT rt‖rt‖2

rt

)+ f +

µ⊕‖rt‖3

rt

⇒ r = − µ⊕‖rt‖3

(r − 3

rT rt‖rt‖2

rt

)+ f . (A–3)

This relative motion model in equation (A–3) expressed in the target’s orbital frame is

⇒ r = −n2(

r − 3rT rt‖rt‖2

rt

)+ f − ωo × r − 2ωo × r − ωo × (ωo × r) . (A–4)

where the derivatives are derivates in the target’s rotating orbital frame. The target

position can be expressed in the target’s orbital frame as

rt =

[rt 0 0

]T.

161

where rt is the distance from the Earth’s center to the target. In addition, the angular

velocity of the target’s orbital frame is expressed in the target’s orbital frame as

ωo =

[0 0 ω

]T,

where ω is the target’s orbital rate. Letting the relative position vector be defined as

r =

[x y z

]T,

then the cross product terms are simplified as

ωo × r =

[−ωy ωx 0

]Tωo × r =

[−ωy ωx 0

]Tωo × (ωo × r) =

[−ω2x ω2y 0

]Tr − 3

rT r

‖r‖2 rt =

[−2x y z

]TFrom the orbital angular momentum relationship

r 2t ω = h ,

the simplification can be made

µ⊕‖rt‖3

=µ⊕

h32

ω32 = kω

32 ,

where

k =µ

h32

162

is a constant. As a result, equation (A–4) simplifies to

x − 2kω32 x − ω2x − ωy − 2ωy = fx

y + kω32 y − ω2y + ωx + 2ωx = fy (A–5)

z + kω32 z = fz ,

where fx , fy , fz are the components of the control acceleration f in the target’s orbital

frame.

A.1.1 Clohessy-Wiltshire-Hill Equation

Assuming the target is in a circular orbit, then

ω = 0

k = ω12 .

Simplifying equation (A–5) results in the Clohessy-Wiltshire-Hill (CWH) equation which

describes the relative translational motion between two spacecraft [39, 55]

x − 3n2x − 2ny = fx

y + 2nx = fy (A–6)

z + n2z = fz ,

where n = ω is the mean motion of the target’s orbital frame. Equation (A–6) can be

written in state space form r

v

=

0 I

A21 A22

r

v

+

0

I

u ,

163

where

r =

[x y z

]T,

v =

[x y z

]T,

u =

[fx fy fz

]T,

and A21, A22 are blocks of the state matrix and are defined as

A21 =

3n2 0 0

0 0 0

0 0 −n2

A22 =

0 2n 0

−2n 0 0

0 0 0

.

A.1.2 Yamanaka-Ankerson-Tschauner-Hempel Equation

By relaxing the circular orbit assumption, a different set of equations can be

obtained. [98, 99] First, the independent variable is changed from time to the true

anomaly of the target (i.e., θ). Through the chain rule, the time derivative of an arbitrary

scalar parameter a now becomes

da

dt=da

dθ

dθ

dt

⇒ dadt

= ωa′ ,

164

and the second time derivative becomes

d2a

dt2=dω

dta′ + ω

da′

dt

⇒ d2a

dt2=dω

dθ

dθ

dta′ + ω

da′

dθ

dθ

dt

⇒ d2a

dt2= ωω′a′ + ω2a′′ .

Using these relationships, equation (A–5) can be written as

ω2x ′′ + ωω′x ′ − 2kω32 x − ω2x − ωω′y − 2ω2y ′ = fx

ω2y ′′ + ωω′y ′ + kω32 y − ω2y + ωω′x + 2ω2x ′ = fy (A–7)

ω2z ′′ + ωω′z ′ + kω32 z = fz ,

Recall that the position magnitude in terms of the true anomaly can be written as

rt =p

1 + e cos θ,

where p is the semi-latus rectum of the target’s orbit and e is the eccentricity of the

target’s orbit. [40] Similarly, from the conservation of orbital angular momentum, the

semi-latus rectum can be written as

p =h2

µ⊕. (A–8)

As a result, the orbital rate of the target can be written as

ω =h

r 2t=h

p2(1 + e cos θ)2 = k2(1 + e cos θ)2 , (A–9)

and the derivative of the orbital rate with respect to the true anomaly is

ω′ = −2k2e sin θ(1 + e cos θ) . (A–10)

Substituting equation (A–9) and equation (A–10) into equation (A–7) and dividing by ω2

yields the Yamanaka-Ankerson-Tschauner-Hempel (YATH) equations which describe the

165

relative translational motion between two spacecraft

x ′′ −(

1 +2

1 + e cos θ

)x +

2e sin θ

1 + e cos θy − 2e sin θ

1 + e cos θx ′ − 2y ′ =

1

k4(1 + e cos θ)4fx

y ′′ − 2e sin θ

1 + e cos θx −

(1− 1

1 + e cos θ

)y + 2x ′ − 2e sin θ

1 + e cos θy ′ =

1

k4(1 + e cos θ)4fy

z ′′ +1

1 + e cos θz − 2e sin θ

1 + e cos θz ′ =

1

k4(1 + e cos θ)4fz .

These set of equations can be written in state space formr′

v′

=

0 I

A21 A22

r

v

+1

k4(1 + e cos θ)4

0

I

f

where

r =

[x y z

]Tv =

[x ′ y ′ z ′

]Tf =

[fx fy fz

]Tand A21, A22 are blocks of the state matrix and are defined as

A21 =1

1 + e cos θ

3 + e cos θ −2e sin θ 0

2e sin θ e cos θ 0

0 0 −1

A22 =1

1 + e cos θ

2e sin θ 2 0

−2 2e sin θ 0

0 0 2e sin θ

.

A.2 Relative Orientation

A small angle approximation (SAA) can be used to describe the relative orientation

of the chaser with respect to the target. The underlying assumptions here are small

166

angular displacements between the two body frames and a passively tumbling target

target with constant angular velocity ωt relative to an inertial frame. [65] Figure A-1

illustrates the small angular displacement assumption, where bt,1, bt,2, bt,3 is the basis

associated with the target’s body frame Bt and bc,1, bc,2, bc,3 is the basis associated

with the chaser’s body frame Bc . Using an antisymmetric Euler sequence to relate the

orientation of the target body frame to the chaser body frame, the rotation matrix to

relate two reference frames is approximated as

CBcBt ≈ I−α× ,

where

α =

[α1 α2 α3

]Tand α1,α2,α3 are the angles associated with the rotations in the antisymmetric Euler

sequence. [65] As a result, the relative angular velocity is α and the angular velocity of

the chaser body frame relative to an inertial frame is approximated as

ωc = α+ (I−α×)ωt .

€

bc,1

€

bt,1

€

bc,2

€

bt,2

€

bc,3

€

bt,3

€

α3

€

α2

€

α1

Figure A-1. Relative orientation between chaser and target

167

Applying Euler’s second law, the governing equations for small angular displacements

of the chaser body frame relative to the target body frame is

J(α− α×ωt

)+(α+

[(I−α×

)ωt])×

J(α+

[I−α×

]ωt)

= τ , (A–11)

where J is the inertia matrix of the chaser. Considering only the first order terms,

equation (A–11) reduces to

Jα =[ωtω

Tt J− ω×t Jω×t −

(ωTt Jωt

)I]α+

[(Jωt)

× − ω×t J− Jω×t]α− ω×t Jωt + τ .

(A–12)

Equation (A–12) can be written in state space formαα

=

0 I

A21 A22

αα

+

0

J−1

u ,

where u = τ − ω×t Jωt and A21, A22 are blocks of the state matrix and are defined as

A21 = J−1[ωtω

Tt J− ω×t Jω×t −

(ωTt Jωt

)I]

A22 = J−1[(Jωt)

× − ω×t J− Jω×t]

.

It is interesting to note that a constant force term appears in this set of equations due to

the constant tumbling rate of the target.

168

APPENDIX BTWO POINT BOUNDARY VALUE PROBLEM SOLUTION

The two point boundary value problem (TPBVP) is stated as: “Given a dynamical

system x = f(x, u, t), an initial condition x(t0) = x0, and a desired final condition

x(tf ) = xf obtained at time t = tf , determine the control action u(t) that satisfies these

conditions.” The problem of interest is to determine an initial change in velocity ∆xi to

reach a final position xf in time T with arbitrary final velocity. Solving a TPBVP can

be cumbersome, but this is not the case for linear systems. Since the relative motion

equations are linear, they can be written using a block form of the state transition matrix

(STM) representation as [71]x(t)

x(t)

=

Φ11(t) Φ12(t)

Φ21(t) Φ22(t)

x0

x0

+

∫ tt0

Φ11(t − ε) Φ12(t − ε)

Φ21(t − ε) Φ22(t − ε)

Bu(ε) dε . (B–1)

Thus, the solution of the TPBVP of interest is obtained by letting

x0 = x0 + ∆xi

and solving a system of algebraic equations for ∆xi .

The TPBVP solution is obtained for both the CWH equation and SAA equation.

Since the SAA equation has a constant forcing term, the associated TPBVP problem

includes a general constant forcing term u = −ω×t Jωt . Also, note that for both relative

motion equations the input matrix is of the form

B =

0

B2

.

Thus, considering the first row of equation (B–1), the initial change in velocity required to

reach a final position xf in time T is solved for as

∆xi = Φ−112 (T ) [xf − xt −Φ11(T )x0]− x0 , (B–2)

169

where

xt =

∫ Tt0

Φ12(T − t)B2u dt .

This is for the application of interception since the only condition is for x→ xf as t → T .

Substituting equation (B–2) into equation (B–1) and considering the second row of

equation (B–1) yields the velocity profile

xd(t) = Φ12(t)x0 + Φ22(t)[x0 + Φ−112 (T ) (xf − xt −Φ11(T )x0)− x0

]...

+

∫ tt0

Φ22(T − ε)B2u dε

⇒ xd(t) =[Φ21(t)−Φ22(t)Φ−112 (T )Φ11(T )

]x0 + Φ22(t)Φ−112 (T ) [xf − xt ] ...

+

∫ tt0

Φ22(T − ε)B2u dε . (B–3)

Recall the STM property [71]

Φ(t1)Φ(t2) = Φ(t1 + t2) .

Using the block form of the STM and letting t1 = T and t2 = −T , the (1, 2) block of this

STM property is

Φ11(T )Φ12(−T ) + Φ12(T )Φ22(−T ) = 0 .

Multiplying on the left by Φ−112 (T ) and multiplying on the right by Φ−112 (−T ), the equation

reduces to

Φ−112 (T )Φ11(T ) = −Φ22(−T )Φ−112 (−T ) . (B–4)

170

Using the same STM property and letting t1 = t and t2 = −T , the (2, 2) block of the

STM property for this case is

Φ21(t)Φ12(−T ) + Φ22(t)Φ22(−T ) = Φ22(t − T )

⇒ Φ22(t)Φ22(−T ) = Φ22(t − T )−Φ21(t)Φ12(−T ) . (B–5)

Substituting equation (B–4) into equation (B–3) results in equation(B–3)

xd(t) =[Φ21(t)−Φ22(t)Φ−122 (−T )Φ−112 (−T )

]x0 + Φ22(t)Φ−112 (T ) [xf − xt ] ...

+

∫ tt0

Φ22(T − ε)B2u dε .

Substituting equation (B–5) into this equation results in

xd(t) =[Φ21(t)− (Φ22(t − T )−Φ21(t)Φ12(−T )) Φ−112 (−T )

]x0 + Φ22(t)Φ−112 (T ) [xf − xt ] ...

+

∫ tt0

Φ22(T − ε)B2u dε

⇒ xd(t) = Φ22(t − T )Φ−112 (−T )x0 + Φ22(t)Φ−112 (T ) [xf − xt ] +

∫ tt0

Φ22(T − ε)B2u dε .

(B–6)

This velocity profile gives the velocity history of the trajectory from the initial condition to

the final position.

Equation (B–6) can also be used to define the velocity change ∆xf needed at

position xf so that the velocity at time T is zeroed out. This velocity change is the

negative of the velocity profile at time t = T .

∆xf = −Φ−112 (−T )x0 −Φ22(T )Φ−112 (T ) [xf − xt ]−∫ Tt0

Φ22(T − t)B2u dt

This is for the application of rendezvous since x→ xf , x→ 0 as t → T .

Obtaining these two velocity changes is particularly useful when considering a

minimum control effort OCP. The solution to the minimum control effort optimal control

171

problem (OCP) can be approximated as the two burn solution, with one burn at the

initial time and one burn at the final time. [72] Thus, the cost for this problem is also

approximated as

J ≈ ‖∆xi‖1 + ‖∆xf ‖1 .

It should be noted that the matrix Φ12(T ) can be singular for certain values of

T . To illustrate this, two interception examples are performed using the CWH and

SAA equations. The parameters used for the CWH example are in Table B-1, and the

parameters for the SAA example are in Table B-2.

Table B-1. Relative orientation parametersParameter Value Units

n 0.001 rad/sr0

[500 500 500

]T mv0

[0 0 0

]T m/srf

[0 0 0

]T m

Table B-2. Relative orientation parametersParameter Value Units

J

300 0 00 100 00 0 200

kg ·m2

ωt[0.1 0.1 0.1

]T rad/sα0

[0.1 0.1 0.1

]T radα0

[0 0 0

]T rad/sαf

[0 0 0

]T rad

Figure B-1A shows a set of TPBVP trajectories for the CWH equation with different

values of T . Figure B-1B shows a plot of ‖∆vi‖1 for different values of T and depicts

the values where Φ12(T ) becomes singular (i.e., the vertical asymptotes). For this

example, the matrix Φ12(T ) becoming singular physically signifies an infinite impulse

that is pointed directly at the desired position or in the opposite direction.

172

Figure B-1C shows a set of TPBVP trajectories for the SAA equation with different

values of T . Figure B-1D shows a plot of ‖∆αi‖1 for different values of T and depicts

the values where Φ12(T ) becomes singular (i.e., the vertical asymptotes). For this

example, the matrix Φ12(T ) becoming singular physically signifies an infinite impulse to

reach the desired orientation.

0

100

200

300

400

500

−200

0

200

400

−1000

0

1000

2000

x (m)y (m)

z(m

)

A CWH TPBVP trajectories

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

T (ksec)

‖∆vi‖

1(m

/s)

B CWH TPBVP impulse magnitudes

0.050.1

0.150.2

0

0.1

0.2

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

α1 (rad)α2 (rad)

α3(rad)

C SAA TPBVP trajectories

0 10 20 30 40 50 600

1

2

3

4

5

6

7

8

9

10

T (ksec)

‖∆α

i‖1(rad/s)

D SAA TPBVP impulse magnitudes

Figure B-1. Two point boundary value problem examples

173

APPENDIX CHOMOTOPY WITH RICCATI EQUATION - L(S) = S

The derivation for solving the nonlinear operator

N(S) = S + Q− SBR−1BTS + ATS + SA = 0 , S(tf ) = Sf , (C–1)

using the Homotopy Continuation (HC) method is presented in this Appendix. First, the

convex homotopy is defined as

H(S, p) = (1− p) [L(S)− L(Y0)]− phΨ(t) [N(S)] = 0 , (C–2)

where p ∈ ν ∈ R | 0 ≤ ν ≤ 1 is the embedding parameter, h ∈ R− 0 is the auxiliary

parameter, Ψ(t) 6= 0 ∀ t is the auxiliary function, Y0 is an initial guess, and

L(S) = S .

Next, the Maclaurin Series Expansion (MSE) of S is obtained as

S = S0 + pS1 + p2S2 + · · · , (C–3)

Substituting equation (C–3) into equation (C–2) yields,

H(S0, S1, S2, ... , p) = (1− p)[L(S0 + pS1 + p2S2 + · · · )− L(Y0)

]− ...

phΨ(t)N(S0 + pS1 + p2S2 + · · · ) (C–4)

The operator L(S) is easily decomposed as

L(S0 + pS1 + p2S2 + ... ) = L(S0) + pL(S1) + p2L(S2) + · · · , (C–5)

since it is a linear operator. The operator N(S) defined in equation (C–1) is nonlinear

and must be decomposed manually. Direct substitution of the MSE into equation (C–1)

174

results in

N(S0, S1, S2, ... , p) =[S0 + pS1 + p2S2 + · · ·

]+ Q− ...[

S0 + pS1 + p2S2 + · · ·]

BR−1BT[S0 + pS1 + p2S2 + · · ·

]+ ...

AT[S0 + pS1 + p2S2 + · · ·

]+[S0 + pS1 + p2S2 + · · ·

]A = 0 .

The result is rewritten such that coefficients are grouped for the different powers of the

embedding parameter,

N(S) = G0 + pG1 + p2G2 + · · · (C–6)

where

G0 = S0 + Q− S0BR−1BTS0 + ATS0 + S0A

G1 = S1 − S0BR−1BTS1 − S1BR−1BTS0 + ATS1 + S1A

G2 = S2 − S0BR−1BTS2 − S1BR−1BTS1 − S2BR−1BTS0 + ATS2 + S2A

G3 = S3 − S0BR−1BTS3 − S1BR−1BTS2 − S2BR−1BTS1 − S3BR−1BTS0 + ATS3 + S3A .

...

Equation (C–5) and equation (C–6) are substituted into equation (C–4) to obtain,

H(S0, S1, S2, ... , p) = (1− p)[L(S0) + pL(S1) + p2L(S2) + · · ·

]− (1− p)L(Y0)− ...

phΨ(t)[G0 + pG1 + p2G2 + · · ·

]= 0

The coefficients for the different powers of the embedding parameter are grouped, which

results in

175

p0 terms : L(S0)− L(Y0)

p1 terms : L(S1)− hΨG0 − L(S0) + L(Y0)

p2 terms : L(S2)− hΨG1 − L(S1)

p3 terms : L(S3)− hΨG2 − L(S2) .

...

The trend for obtaining the subsequent group of terms is the same. The coefficients are

individually set to zero to solve for the parameters S0, S1, S2, ... . The result is a system

of differential equations. For the derivation presented in this Appendix, the auxiliary

function used is Ψ(t) = I, however, this function is free to choose.

The equation for the zeroth order term suggests

L(S0)− L(Y0) = 0

⇒ S0 − Y0 = 0 .

The most easily obtainable solution is to let S0 = Y0 where Y0 is free to choose. The

chosen parameter is Y0 = C0 = Sf since this would satisfy the equation

L(Y0) = 0 ,

and the boundary condition Y0(tf ) = Sf . Thus, the solution for the zeroth parameter is

S0 = C0 = Sf .

The equation for the first order term suggests

L(S1)− hG0 − L(S0) + L(Y0) = 0

⇒ S1 − hG0 = 0 .

176

Note that the term G0 only depends on S0 which has already been solved for. To solve

for the parameter S1, the homogeneous solution is first obtained

S1,h = C1 ,

where C1 is a constant. The particular solution is determined by employing the variation

of parameters technique. That is, the integration constant is assumed to be time

dependent (i.e., C1 = C1(t, tf )) and the homogeneous solution is substituted into the

original differential equation. Thus, the resulting differential equation for the particular

solution is

C1 = hG0 .

Notice that this equation only depends on S0, which has already been solved for. The

solution for the constant is

C1(t, tf ) = h

∫ ttf

G0 dε .

Also, note that C1(tf , tf ) = 0 in order to maintain the boundary condition of S(tf ) = Sf .

Thus, the solution for the first parameter is

S1 = C1(t, tf ) .

The equation for the second order term suggests

L(S2)− hG1 − L(S1) = 0

⇒ S2 − hG1 − S1 = 0 .

Note again that the term G1 only depends on S0 and S1 which have already been solved

for. It should be noted that the subsequent differential equations to be solved are of the

177

same form

Si+1 − hGi − Si = 0 for i = 1, 2, 3, ... .

Thus, for the remaining derivation is done for the (i + 1)th term. The homogeneous

solution of Si+1 is first found and is of the form

Si+1,h = Ci+1 for i = 1, 2, 3, ... ,

where Ci+1 is a constant. The variation of parameters technique is again employed

to solve for the particular solution. The resulting differential equation for the particular

solution is

Ci+1 = Ci + hGi for i = 1, 2, 3, ... .

This equation is integrated to obtain the solution

Ci+1(t, tf ) = Ci + h

∫ ttf

Gi dε for i = 1, 2, 3, ... .

Note again that Ci+1(tf , tf ) = 0 to maintain the boundary condition of S(tf ) = Sf . The

solution for the (i + 1)th parameter is

Si+1 = Ci+1(t, tf ) for i = 1, 2, 3, ... .

As a result, the k th order approximation of the homotopy solution is given by

S(k) =

k∑i=0

Si .

178

APPENDIX DHOMOTOPY WITH TRANSFORMED RICCATI EQUATION - L(X) = X

The derivation for solving the linear time varying (LTV) differential equation

N(X) = X−HtX = 0 , X(tf ) = Xf ,

using the Homotopy Continuation (HC) method is presented in this Appendix. First, the

convex homotopy is defined as

H(X, p) = (1− p) [L(X)− L(Y0)]− phΨ(t) [N(X)] = 0 , (D–1)

where p ∈ ν ∈ R | 0 ≤ ν ≤ 1 is the embedding parameter, h ∈ R− 0 is the auxiliary

parameter, Ψ(t) 6= 0 ∀ t is the auxiliary function, Y0 is an initial guess, and

L(X) = X .

Next, the Maclaurin Series Expansion (MSE) of X is obtained as

X = X0 + pX1 + p2X2 + · · · , (D–2)

Substituting equation (D–2) into equation (D–1) yields

H(X0, X1, X2, ... , p) = (1− p)[L(X0 + pX1 + p2X2 + · · · )− L(Y0)

]− ...

phΨ(t)N(X0 + pX1 + p2X2 + · · · ) . (D–3)

Note that both L(X) and N(X) are linear operators and are easily decomposed as

L(S0 + pS1 + p2S2 + ... ) = L(S0) + pL(S1) + p2L(S2) + · · · (D–4)

N(S0 + pS1 + p2S2 + ... ) = N(S0) + pN(S1) + p2N(S2) + · · · . (D–5)

179

Equation (D–4) and equation (D–5) are substituted into equation (D–3) to obtain

H(X0, X1, X2, ... , p) = (1− p)[L(X0) + pL(X1) + p2L(X2) + · · ·

]− (1− p)L(Y0)− ...

phΨ(t)[N(X0) + pN(X1) + p2N(X2) + · · ·

]= 0 .

The coefficients for the different powers of the embedding parameter are grouped which

results in

p0 terms : L(X0)− L(Y0)

p1 terms : L(X1)− hΨ(t)N(X0)− L(X0) + L(Y0)

p2 terms : L(X2)− hΨ(t)N(X1)− L(X1) (D–6)

p3 terms : L(X3)− hΨ(t)N(X2)− L(X2) .

...

The trend for obtaining the subsequent group of terms is the same. The coefficients are

individually set to zero to solve for the parameters X1, X2, X2, ... . The result is a system

of differential equations. For the derivation presented in this Appendix, the auxiliary

function used is Ψ(t) = I, however, this function is free to choose.

The equation for the zeroth order term suggests

L(X0)− L(Y0) = 0

⇒ X0 − Y0 = 0 .

The most easily obtainable solution is to let X0 = Y0 where Y0 is free to choose. The

chosen parameter is Y0 = C0 = Sf since this would satisfy the equation

L(Y0) = 0 ,

and the boundary condition Y0(tf ) = Xf . Thus, the solution for the zeroth parameter is

S0 = C0 = Xf . (D–7)

180

The equation for the first order term suggests

L(X1)− hN(X0)− L(X0) + L(Y0) = 0

⇒ X1 − h[X0 −HtX0

]= 0 .

To solve for the parameter X1, the homogeneous solution is first obtained

X1,h = C1 ,

where C1 is a constant. The particular solution is determined by employing the variation

of parameters technique. That is, the integration constant is assumed to be time

dependent (i.e., C1 = C1(t, tf )) and the homogeneous solution is substituted into the

original differential equation. Thus, the resulting differential equation for the particular

solution is

C1 = −hHtC0 ,

Notice that this equation only depends on S0, which has already been solved for. The

solution for the constant is

C1(t, tf ) = −h∫ ttf

HtC0 dε .

Note that C1(tf , tf ) = 0 in order to maintain the boundary condition of X(tf ) = Xf . Thus,

the solution for the first parameter is

X1 = C1(t, tf ) .

The equation for the second order term suggests

L(X2)− hN(X1)− L(X1) = 0

⇒ X2 − h[X1 −HtX1

]− X1 = 0 .

181

Note again that the equation only depends on X0 and X1 which have already been

solved for. It should also be noted that the subsequent differential equations to be solved

are of the same form,

Xi+1 − h[Xi −HtXi

]− Xi = 0 for i = 1, 2, 3, ... .

Thus the remaining derivation is done for the (i + 1)th term. The homogeneous solution

of Xi+1 is first found and is of the form

Xi+1,h = Ci+1 for i = 1, 2, 3, ... ,

where Ci+1 is a constant. The variation of parameters technique is again employed

to solve for the particular solution. The resulting differential equation for the particular

solution is

Ci+1 = (h + 1)Ci − hHtCi for i = 1, 2, 3, ... .

This equation is integrated to obtain the solution

Ci+1(t, tf ) = (h + 1)Ci − h∫ tt0

HtCi dε, for i = 1, 2, 3, ... .

Notice again that Ci+1(tf , tf ) = 0 to maintain the boundary condition of X(tf ) = Xf .

Therefore the solution for the (i + 1)th parameter is

Xi+1 = Ci+1(t, tf ) for i = 1, 2, 3, ... .

As a result, the k th order approximation of the homotopy solution is given by

X(k) =

k∑i=0

Xi .

182

APPENDIX EHOMOTOPY WITH TRANSFORMED RICCATI EQUATION - L(X) = X− X

The derivation for solving the linear time varying (LTV) differential equation

N(X) = X−HtX = 0 , X(tf ) = Xf ,

using the Homotopy Continuation (HC) method is presented in this Appendix. First, the

convex homotopy in equation (D–1) is defined with

L(X) = X− X .

Next, the Maclaurin Series Expansion (MSE) of X in equation (D–2) is obtained.

Substituting equation (D–2) into equation (D–1) yields equation (D–3). Since L(X) and

N(X) are linear operators, they can easily be decomposed as shown in equation (D–4)

and equation (D–5), respectively. As a result, the coefficients for the different powers of

the embedding parameter are grouped and result in the same form as equation (D–6).

For the derivation presented in this Appendix, the auxiliary function used is Ψ(t) = I,

however, this function is free to choose.

The equation for the zeroth order term suggests

L(X0)− L(Y0) = 0

⇒[X0 − Y0

]−Hc [X0 − Y0] = 0 .

The most easily obtainable solution is to let X0 = Y0 where Y0 is free to choose. The

chosen parameter is

Y0 = exp((t − tf )I)C0 = exp((t − tf )I)Xf ,

since this would satisfy the equation

L(Y0) = 0 ,

183

and the boundary condition Y0(tf ) = Xf . Thus, the solution for the zeroth parameter is

X0 = exp((t − tf )I)C0 = exp((t − tf )I)Xf .

The equation for the first order terms suggests,

L(X1)− hN(X0)− L(X0) + L(Y0) = 0

⇒[X1 −HcX1

]− h

[X0 −HtX0

]= 0 .

To solve for the parameter X1, the homogeneous solution is first obtained

X1,h = exp((t − tf )I)C1 ,

where C1 is a constant. The particular solution is determined by employing the variation

of parameters technique. That is, the integration constant is assumed to be time

dependent (i.e., C1 = C1(t, tf )) and the homogeneous solution is substituted into the

original differential equation. Thus, the resulting differential equation for the integration

constant is

exp((t − tf ))C1 = h [exp((t − tf )I)C0 −Ht exp((t − tf )I)C0]

⇒ C1 = h [I−Ht ] C0 .

Note that the matrix exponential term is a diagonal term, thus it commutes with the

matrix multiplication operation. The solution for the constant is

C1(t, tf ) = h

∫ ttf

[I−Ht ] C0 dε .

Note that C1(tf , tf ) = 0 in order to maintain the boundary condition of X(tf ) = Xf . Thus,

the solution for the first parameter is

X1 = exp((t − tf )I)C1(t, tf ) .

184

The equation for the second order term suggests

L(X2)− hN(X1)− L(X1) = 0

⇒[X2 − X2

]− h

[X1 −HtX1

]−[X1 − X1

]= 0 .

It should be noted that the subsequent differential equations to be solved are of the form

[Xi+1 − Xi+1

]− h

[Xi −HtXi

]−[Xi − Xi

]= 0 for i = 1, 2, 3, ... .

The remaining derivation are done for the (i + 1)th term. The homogeneous solution of

Xi+1 is first found and is of the form

Xi+1,h = exp((t − tf )I)Ci+1 for i = 1, 2, 3, ... ,

where Ci+1 is a constant. The variation of parameters technique is again employed

to solve for the particular solution. The resulting differential equation for the particular

solution is

exp((t − tf )I)Ci+1 = exp((t − tf )I)Ci + h[exp((t − tf )I)

(Ci + Ci

)−Ht exp((t − tf )I)Ci

]⇒ Ci+1 = (1 + h)Ci + h [I−Ht ] Ci for i = 1, 2, 3, ... .

This equation is integrated to obtain the solution

Ci+1(t, tf ) = (1 + h)Ci + h

∫ ttf

[I−Ht ] Ci dε for i = 1, 2, 3, ... .

Notice again that Ci+1(tf , tf ) = 0 to maintain the boundary condition of X(tf ) = Xf .

Therefore, the solution for the (i + 1)th parameter is

Xi+1 = exp((t − tf )I)Ci+1 for i = 1, 2, 3, ... .

As a result, the k th order approximation of the homotopy solution is given by

X(k) =

k∑i=0

Xi .

185

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BIOGRAPHICAL SKETCH

Josue David Munoz was born in Guatemala City, Guatemala in 1982. He moved

to Miami, Florida at the ripe age of five years old and was raised there. He received

his Bachelors of Science in aerospace engineering from the University of Florida in

December 2005 and graduated Cum Laude. He was then accepted to the Ph.D. direct

program at the University of Florida and obtained his Masters of Science in aerospace

engineering in December 2008. He also had the honor of being awarded the South East

Alliance for Graduate Education of the Professoriate Fellowship as well as the Science,

Math, and Research Transformation Scholarship.

He had the pleasure of being a graduate teaching assistant as well as a graduate

research assistant, working on projects for agencies like Defense Advanced Research

Agency Projects and the Lockheed Martin Corporation. He was also Space Scholar at

the Air Force Research Lab/Space Vehicles Directorate for the Summers of 2009 and

2010. He was also able to be part of the Student Temporary Employment Program at

the Air Force Research Lab for the Fall 2009 semester, where he was able to contribute

with his research. He was a member of the Space Systems Group and Small Satellite

Design Club at the University of Florida, and is a member of the American Institute of

Aeronautics and Astronautics, and American Astronautics Society.

194